text:
Brief Synopsis
A beautiful young woman stands trial for poisoning her lover.
Ann Todd
Norman Woland
Ivan Desny
Leslie Banks
Barbara Everest
Madeleine (1950) - Many Eyes Are Upon Her
Madeleine (1950) - We Are Quite Alone
Theatrical Aspect Ratio
Based on the true story of Madeleine Smith, a young Glasgow woman from a wealthy family, she stood trial in 1857 for the murder of her lover, Emile L'Angelier. Her trial was much publicized in the newspapers of the day and was labeled "the trial of the century."
Susan Stranks
Patricia Raine
Elizabeth Sellars
Jean Cadell
Stanley Haynes
Nicholas Phipps
George Pollack
Madeleine (1950) - Many Eyes Are Upon Her At a Glasgow society ball, Mrs. Smith (Barbara Everest) chats while her daughter (Ann Todd, title character) dances first with her father (Leslie Banks) then with Minnoch (Norman Wooland), spurned lover L'Angelier (Ivan Desny) observing, in director David Lean's fact-based Madeleine, 1950.
Madeleine (1950) - We Are Quite Alone Affluent Ann Todd (title character) has lured her lover from Glasgow (Ivan Desny) to her family's country estate, joining him outside the grounds for an encounter overlooking the neighboring village, Todd's husband, the director David Lean, engaging in some exhilarating cutting, in Madeleine, 1950, based on a famous 1857 murder case.
The case of Madeleine Smith had been the scandal of the day in 1857 Glasgow, Scotland. Twenty-one year old Smith was accused of killing her lover for not returning her love letters when she became engaged to another man. While there was no actual evidence of her crime and her guilt was not proven in the courtroom, it was widely believed that she was indeed guilty.
Both the film Madeleine (1950) and the real-life events on which it was based were surrounded in scandal. Smith left her lover to marry another man, the actress playing her, Ann Todd, had just divorced her husband to marry her lover, Madeleine director David Lean.
The Smith case had been made into the play Dishonored Lady by Edward Sheldon and Margaret Ayer Barnes in which Katharine Cornell had starred on Broadway in 1928. The film adaptation of the play had been filmed only three years before in 1947 starring Hedy Lamarr. Todd had played Madeleine Smith on stage in The Rest Is Silence and desperately wanted to make a film. Lean was less enthusiastic, "I had just married Ann Todd and she begged me to direct it. It was a miserable film, one of the most difficult I've ever made. Something didn't fit. I don't know what," Lean later recalled. As Gene D. Phillips wrote in his book, Beyond the Epic: The Life & Films of David Lean, the problem might have been personal. "While Lean was filming a beach scene for Madeleine, as the movie was to be called, a small plane interrupted a shot by swooping down on the terrified cast and crew and flying very low along the shore. Lean angrily shook his fist and shouted imprecations at the pilot, whom he did not recognize. But Todd did; it was [Nigel] Tangye [her recently divorced husband], whom Lean had never met. She never told Lean since he had said that he was going to report the pilot to the local flying club if he learned his identity. This nasty prank was, presumably, Tangye's parting shot. But Todd told me that she did not hold it against him. Nigel was an absolute saint, she said, and he just was not himself for some time after the ordeal of the divorce." Madeleine was shot at Pinewood Studios in the summer of 1949, shortly after the Lean-Todd marriage and locations were in Cornwall and Scotland. It was not an idyllic honeymoon, as Phillips wrote, "Todd fell into the same pattern of behavior with Lean that had driven Ronald Neame to distraction when he was briefly directing The Passionate Friends [1949]. That is, she incessantly attempted to second-guess the director about how each scene should be played. She engaged Lean in lengthy discussions during rehearsals in which she endeavored to psychoanalyze her character, and these endless squabbles put the movie behind schedule. To make matters worse, there was an electricians' strike at the studio in August, which caused further delays."
Todd's obsession with the Smith case led to her owning some letters that Smith had written to her lover Emile L'Angelier, and an ivory-handled parasol that Smith brought to her trial, and which Todd herself had used in a courtroom scene in The Rest Is Silence. Lean would add to the verisimilitude of the film by shooting on location at Smith's own house at 7 Blythswood Square in Glasgow; and by using court transcripts in the trial scenes in the last third of the movie', including the final summation by Smith's attorney.
Madeleine was not successful when it was released in London in February 1950. Noel Coward summed up the problem that audiences had with it, "I don't think you can end the film not knowing whether she did or didn't kill him. Somehow you've got to tip the scales one way or the other."
Producer: Stanley Haynes
Director: David Lean
Screenplay: Stanley Haynes, Nicholas Phipps
Cinematography: Guy Green
Music: William Alwyn
Film Editing: Clive Donner, Geoffrey Foot
Cast: Ann Todd (Madeleine Smith), Norman Wooland (William Minnoch), Ivan Desny (Emile L'Anglier), Leslie Banks (James Smith), Barbara Everest (Mrs. Smith), Elizabeth Sellars (Christina Hackett), Patricia Raine (Bessie Smith), Eugene Deckers (Thuau), Andre Morell (Defending Counsel), Barry Jones (Prosecuting Counsel), Susan Stranks (Janet Smith).
BW-115m.
by Lorraine LoBianco
Beyond the Epic: The Life & Films of David Lean by Gene D. Phillips
Camera Journal: Films and Filmmakers by Paul Sutton, September 14, 2007
The case of Madeleine Smith had been the scandal of the day in 1857 Glasgow, Scotland. Twenty-one year old Smith was accused of killing her lover for not returning her love letters when she became engaged to another man. While there was no actual evidence of her crime and her guilt was not proven in the courtroom, it was widely believed that she was indeed guilty. Both the film Madeleine (1950) and the real-life events on which it was based were surrounded in scandal. Smith left her lover to marry another man, the actress playing her, Ann Todd, had just divorced her husband to marry her lover, Madeleine director David Lean. The Smith case had been made into the play Dishonored Lady by Edward Sheldon and Margaret Ayer Barnes in which Katharine Cornell had starred on Broadway in 1928. The film adaptation of the play had been filmed only three years before in 1947 starring Hedy Lamarr. Todd had played Madeleine Smith on stage in The Rest Is Silence and desperately wanted to make a film. Lean was less enthusiastic, "I had just married Ann Todd and she begged me to direct it. It was a miserable film, one of the most difficult I've ever made. Something didn't fit. I don't know what," Lean later recalled. As Gene D. Phillips wrote in his book, Beyond the Epic: The Life & Films of David Lean, the problem might have been personal. "While Lean was filming a beach scene for Madeleine, as the movie was to be called, a small plane interrupted a shot by swooping down on the terrified cast and crew and flying very low along the shore. Lean angrily shook his fist and shouted imprecations at the pilot, whom he did not recognize. But Todd did; it was [Nigel] Tangye [her recently divorced husband], whom Lean had never met. She never told Lean since he had said that he was going to report the pilot to the local flying club if he learned his identity. This nasty prank was, presumably, Tangye's parting shot. But Todd told me that she did not hold it against him. Nigel was an absolute saint, she said, and he just was not himself for some time after the ordeal of the divorce." Madeleine was shot at Pinewood Studios in the summer of 1949, shortly after the Lean-Todd marriage and locations were in Cornwall and Scotland. It was not an idyllic honeymoon, as Phillips wrote, "Todd fell into the same pattern of behavior with Lean that had driven Ronald Neame to distraction when he was briefly directing The Passionate Friends [1949]. That is, she incessantly attempted to second-guess the director about how each scene should be played. She engaged Lean in lengthy discussions during rehearsals in which she endeavored to psychoanalyze her character, and these endless squabbles put the movie behind schedule. To make matters worse, there was an electricians' strike at the studio in August, which caused further delays." Todd's obsession with the Smith case led to her owning some letters that Smith had written to her lover Emile L'Angelier, and an ivory-handled parasol that Smith brought to her trial, and which Todd herself had used in a courtroom scene in The Rest Is Silence. Lean would add to the verisimilitude of the film by shooting on location at Smith's own house at 7 Blythswood Square in Glasgow; and by using court transcripts in the trial scenes in the last third of the movie', including the final summation by Smith's attorney. Madeleine was not successful when it was released in London in February 1950. Noel Coward summed up the problem that audiences had with it, "I don't think you can end the film not knowing whether she did or didn't kill him. Somehow you've got to tip the scales one way or the other." Producer: Stanley Haynes Director: David Lean Screenplay: Stanley Haynes, Nicholas Phipps Cinematography: Guy Green Music: William Alwyn Film Editing: Clive Donner, Geoffrey Foot Cast: Ann Todd (Madeleine Smith), Norman Wooland (William Minnoch), Ivan Desny (Emile L'Anglier), Leslie Banks (James Smith), Barbara Everest (Mrs. Smith), Elizabeth Sellars (Christina Hackett), Patricia Raine (Bessie Smith), Eugene Deckers (Thuau), Andre Morell (Defending Counsel), Barry Jones (Prosecuting Counsel), Susan Stranks (Janet Smith). BW-115m. by Lorraine LoBianco SOURCES: Beyond the Epic: The Life & Films of David Lean by Gene D. Phillips Camera Journal: Films and Filmmakers by Paul Sutton, September 14, 2007 The Internet Movie Database Wikipedia.org


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text:
\section{Introduction}
Grassmann graphs and polar Grassmann graphs (not necessarily over finite fields)
are interesting for many reasons
\cite{BC-book,D-book,Pankov-book1,Pankov-book2,Pasini-book, Shult-book}.
For example, they are closely related to buildings of classical types \cite{Tits}.
Also, Grassmann graphs and dual polar graphs (over finite fields)
are classical examples of distance regular graphs \cite{BCN-book}.
Embeddings (not necessarily isomorphic in some cases)
of Grassmann graphs and polar Grassmann graphs
(over division rings) are investigated in
\cite{Pankov-paper1,Pankov-book2,KP,Pankov-paper2}.

All dual polar graphs defined by sesquilinear, quadratic and pseudo-quadratic forms
are naturally isometrically embedded in the corresponding Grassmann graphs.
In this short note we consider the Grassmann graphs and dual polar graphs over
the {\it same finite} field.
We show that, up to graph automorphism, for every dual polar graph
there is the unique isometric embedding in the corresponding Grassmann graph.
This statement is related to the problem formulated in \cite{KMP}.

The author's interest is the general case when
Grassmann graphs and dual polar graphs are
related to different not necessarily finite division rings.
To describe isometric embeddings and  get a result in spirit of \cite[Chapter 3]{Pankov-book2},
we need semilinear embeddings of special type.
This is a topic for a more detailed research.

\section{Result}
Let $V$ be an $n$-dimensional  vector space over a division ring.
Denote by ${\mathcal G}_{k}(V)$ the Grassmannian formed by
$k$-dimensional subspaces of $V$.
The corresponding {\it Grassmann graph} $\Gamma_{k}(V)$ is
the graph whose vertex set is ${\mathcal G}_{k}(V)$
and two $k$-dimensional subspaces are adjacent vertices of this graph if their intersection is $(k-1)$-dimensional.
If $k=1,n-1$ then any two distinct vertices of $\Gamma_{k}(V)$ are adjacent and we will alway suppose that
$1<k<n-1$. Also, we can suppose that $2k\le n$, since the Grassmann graphs $\Gamma_{k}(V)$ and $\Gamma_{n-k}(V^{*})$ are
isomorphic.

We write $\Pi_{V}$ for the projective space associated to $V$
(the points are the $1$-dimensional subspaces and the lines are defined by the $2$-dimensional subspaces).
Let $\Pi=({\mathcal P},{\mathcal L})$ be a rank $m$ polar space,
see \cite{BC-book, Ueberberg} for the precise definition.
Also, we suppose that the polar space $\Pi$ is embedded in
the projective space $\Pi_{V}$, i.e.
the lines of $\Pi$ are lines of $\Pi_{V}$.
Then $2m\le n$ and all maximal singular subspaces of $\Pi$ can be identified  with some $m$-dimensional subspaces of $V$.
The set of all such $m$-dimensional subspaces is denoted by ${\mathcal G}(\Pi)$.
The corresponding {\it dual polar graph} $\Gamma(\Pi)$
is the restriction of the Grassmann graph $\Gamma_{m}(V)$
to the set of maximal singular subspaces ${\mathcal G}(\Pi)$.

The existence of isometric embeddings of $\Gamma(\Pi)$ in $\Gamma_{k}(V)$
implies that $m\le k$, i.e.
the diameter of the dual polar graph is not greater than the diameter of the Grassmann graph.

\begin{theorem}
If $m\le k$ and $V$ is a vector space over a finite field then,
up to automorphism of the Grassmann graph $\Gamma_{k}(V)$,
there is the unique isometric embedding of $\Gamma(\Pi)$ in $\Gamma_{k}(V)$.
\end{theorem}

\section{Sketch of proof}

Let $U$ be a subspace of $V$ whose dimension is less than $k$.
Denote by $[U\rangle_{k}$ the set of all $k$-dimensional subspaces containing $U$.
If $U$ is a singular subspace for $\Pi$ then
$$[U\rangle:={\mathcal G}(\Pi)\cap [U\rangle_{m}$$
is the set of all maximal singular subspaces containing $U$.

Let $f:{\mathcal G}(\Pi)\to {\mathcal G}_{k}(V)$
be an isometric embedding of $\Gamma(\Pi)$ in $\Gamma_{k}(V)$,
i.e. an injection preserving the distance between vertices.

\begin{lemma}
The image of $f$ is contained in $[U\rangle_{k}$,
where $U$ is a $(k-m)$-dimensional subspace.
\end{lemma}

\begin{proof}
A simple modification of the proof \cite[Lemma 6]{Pankov-paper1}.
\end{proof}

By Lemma 1, our isometric embedding
can be considered as an isometric embedding $g$
of $\Gamma(\Pi)$ in $\Gamma_{m}(W)$, where $W=V/U$.

Let $P$ be a $1$-dimensional subspace of $V$
which is a point of the polar space $\Pi$.
All lines of $\Pi$ containing $P$ form a polar space of rank $m-1$
and the associated dual polar graph is
the restriction of $\Gamma(\Pi)$ to the set $[P\rangle$.
Lemma 1 implies the existence of a $1$-dimensional subspace $q(P)\subset W$
such that
$$g([P\rangle)\subset [q(P)\rangle_{m}.$$
In other words, our isometric embedding induces a certain mapping
$q:{\mathcal P}\to {\mathcal G}_{1}(W)$.
The mapping $q$ is injective (this follows from the fact that $g$ is an isometric embedding).
Next, we establish the following: if $P$ and $Q$ are collinear points of $\Pi$
then
$$g([P+Q\rangle)\subset [q(P)+q(Q)\rangle_{m}$$
which implies that $q$ sends lines of $\Pi$ to subsets in lines of $\Pi_{W}$
(our objects are not necessarily over the same finite field).

However, if $V$ is over a finite field
then $q$ sends every line of $\Pi$ to a line of $\Pi_{W}$.
The restriction of $q$ to every maximal singular subspace of $\Pi$
is a collineation to a certain projective space.
By the Fundamental Theorem of Projective Geometry,
this restriction is induced by a semilinear isomorphism
between the corresponding vector spaces.
Using \cite[Section III.3, f)]{D-book},
we extend $q$ to a collineation of $\Pi_{V'}$ to $\Pi_{W'}$,
where $V'$ is the minimal subspace of $V$ containing $\Pi$
and $W'$ is a subspace of $W$
\footnote{This arguments do not work if Grassmann graphs and dual polar graphs are
related to different not necessarily finite division rings}.
This gives the claim.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

Throughout this paper, we assume that
\begin{empheq}[box = \mybluebox]{equation*}
\text{$\mathcal{H}$ is a real Hilbert space},
\end{empheq}
with inner product $\innp{\cdot,\cdot}$ and induced norm $\|\cdot\|$.

Following Deutsch and Hundal \cite{DH2006II}, we now recall
two notions of angles between two nonempty convex sets:

\begin{definition} {\bf (Dixmier and Friedrichs angle)} {\rm \cite[Definitions~2.3 and 3.2]{DH2006II}} \label{defn:Angles}
Let $C$ and $D$ be nonempty convex sets in $\mathcal{H}$.
The \emph{minimal angle} or \emph{Dixmier angle} between $C$ and $D$ is the angle in
$\left[0, \frac{\pi}{2} \right]$ whose cosine is given by
\begin{empheq}[box = \mybluebox]{equation}
\label{eq:minimalangle}
c_{0} (C,D) := \sup \big\{   \innp{x,y} ~\big| ~ x \in \overline{\cone} (C) \cap \mathbf{B}_{\mathcal{H}}, ~ y \in \overline{\cone} (D) \cap \mathbf{B}_{\mathcal{H}}\big\},
\end{empheq}
where the $\cone$ and $\overline{\cone}$ operator returns
the smallest cone and closed cone containing its argument, respectively, and
where $\mathbf{B}_{\mathcal{H}}$ denotes the closed unit ball in $\mathcal{H}$.
In addition, the \emph{angle} or \emph{Friedrichs angle}
between $C$ and $D$ is the angle in $\left[0, \frac{\pi}{2} \right]$ whose cosine is given by
\begin{empheq}[box = \mybluebox]{equation}
c(C,D) := c_{0} \big( (\cone C) \cap \overline{ C^{\ominus} +D^{\ominus}},
~ (\cone D) \cap  \overline{ C^{\ominus} +D^{\ominus}} \big),
\end{empheq}
and where $C^\ominus := \{ y\in\mathcal{H}~|~ \sup\innp{C,y}\leq 0\}$ is the polar cone of $C$
and $C^\oplus := -C^\ominus$ is the dual cone of $C$.
\end{definition}
The cosine of these angles plays a key role in describing convergence rates
for projection methods such as the standard cyclic projection algorithm.
For a taste of such results, we refer the reader to, e.g.,
\cite{DH2006II}, \cite{DH2008III},
and \cite[Example~9.40]{D2012}.

When $M$ and $N$ are closed linear subspaces of $\mathcal{H}$,
then the cosines of their Dixmier and Friedrichs angles --- which were first studied
in \cite{Dixmier1949} and \cite{Friedrichs1937} --- can be written more succinctly as
\begin{equation*}
	c_{0} (M,N) = \sup \big\{   \innp{x,y} ~\big| ~ x \in M \cap \mathbf{B}_{\mathcal{H}}, ~ y \in N \cap \mathbf{B}_{\mathcal{H}}\big\}
\end{equation*}
and
\begin{equation*}
c(M,N) := c_{0} \big( M \cap (M\cap N)^\perp,
~ N \cap  (M\cap N)^\perp \big),
\end{equation*}
respectively.
The following three key results in this linear setting
were beautifully presented and proved in Deutsch's survey \cite{Deutsch1995} to which
we also refer the reader for interesting historical comments:

\begin{fact} {\bf (Hundal's Lemma)} {\rm (see \cite[Lemma~2.14]{Deutsch1995})} \label{f:Hundal}
Let $M$ and $N$ be closed linear subspaces of $\mathcal{H}$ such that
$c_{0} (M,N) <1$, and  let $X$ be a closed linear subspace such that $M+N\subseteq X \subseteq
\mathcal{H}$. Then
\begin{equation*}
c_{0} (M,N) \leq c_{0} (M^{\perp} \cap X, N^{\perp} \cap X ).
\end{equation*}
\end{fact}

\begin{fact} {\bf (Krein, Krasnoselskii, and Milman)} {\rm (see \cite[Theorem~2.15]{Deutsch1995})} \label{f:KKM}
Let $M$ and $N$ be closed linear subspaces of $\mathcal{H}$ such that
$M\cap N = \{0\}$ and $M+N=\mathcal{H}$. Then
\begin{equation*}
c_{0} (M,N) = c_0(M^\perp,N^\perp).
\end{equation*}
\end{fact}

\begin{fact} {\bf (Solmon)} {\rm (see \cite[Theorem~2.16]{Deutsch1995})} \label{f:Solmon}
Let $M$ and $N$ be closed linear subspaces of $\mathcal{H}$.
Then
\begin{equation*}
c(M,N) = c(M^\perp,N^\perp).
\end{equation*}
\end{fact}

We are now in a position to describe \emph{the aim of this paper:}
{\em We will study the possibility of generalizing \cref{f:Hundal}, \cref{f:KKM}, and \cref{f:Solmon}
from linear subspaces to convex cones.}
While we initially anticipated nice generalizations, it turned out that only
\cref{f:Hundal} appears to admit a natural and nice generalization.
The situation is more complicated for \cref{f:KKM} and \cref{f:Solmon};
our results and examples show that there are ostensibly no ``nice'' conical variants.
Our analysis relies on Deutsch's exposition \cite{Deutsch1995} as well as
our recent work \cite{BOWJMAA}.

The remainder of the paper is organized as follows.
In \cref{sec:aux}, we collect facts useful in subsequent proofs.
The (positive) results concerning Hundal's Lemma (\cref{f:Hundal}) are
presented in \cref{sec:pos}.
The (somewhat negative) results concerning \cref{f:KKM} and \cref{f:Solmon}
are provided in \cref{sec:neg}.

Finally, the notation we employ is standard and follows
\cite{BC2017} and \cite{D2012}.

\section{Auxiliary result}

\label{sec:aux}

In this section, we simply list --- for the reader's convenience ---
several known results that are used in proving our new results
in \cref{sec:pos} and \cref{sec:neg}.

\begin{fact} {\rm \cite[Theorem~4.5]{D2012}} \label{fact:dualcone}
	Let $C$ be a nonempty subset of  $\mathcal{H}$.  Then the following hold:
	\begin{enumerate}
		\item  \label{fact:dualcone:o} $C^{\ominus}$ is a closed convex cone and $C^{\perp}$ is a closed linear subspace.
		\item   \label{fact:dualcone:eq}$C^{\ominus} =(\overline{C})^{\ominus}=\left( \cone(C) \right)^{\ominus}=\left( \overline{\cone}(C) \right)^{\ominus}$.
		\item \label{fact:dualcone:oo} $ C^{\ominus\ominus} = \overline{\cone}(C) $.

		\item  \label{fact:dualcone:closedconvex} If $C$ is a closed convex cone, then $C^{\ominus\ominus} =C$.
		\item  \label{fact:dualcone:linearsubsp} If $C$ is a linear subspace,
		then $C^{\ominus}=C^{\perp}$; if $C$ is additionally closed, then $C=C^{\ominus\ominus}=C^{\perp \perp}$.
	\end{enumerate}
\end{fact}

\begin{fact} {\rm \cite[Lemma~2.5]{BOWJMAA}} \label{lemma:polardual}
	Let $C$  be a nonempty subset  of $\mathcal{H}$.   Then the following statements hold:
	\begin{enumerate}
		\item \label{lemma:polardual:-C} $	(-C)^{\ominus} =- C^{\ominus} =C^{\oplus}$.
		\item \label{lemma:polardual:oplus} $C^{\oplus\oplus}=\overline{\cone} (C)$.

		\item  \label{lemma:polardual:llinear} If $C$ is a linear subspace of $\mathcal{H}$, then $C^{\perp}=C^{\ominus}=C^{\oplus}$.
	\end{enumerate}
\end{fact}

\begin{fact} {\rm \cite[Propositions~6.3 and 6.4]{BC2017} } \label{lemma:K}
Let $K$ be a nonempty convex cone in $\mathcal{H}$.
Then  $K+K=K$.
If $-K \subseteq K$, then $K$ is a   linear subspace.
\end{fact}

\begin{fact}  {\rm \cite[Propositions~6.28]{BC2017}} \label{fact:chara:PK}
	Let $K$ be a nonempty closed convex cone in $\mathcal{H}$, let $x \in \mathcal{H}$, and let
 $p \in \mathcal{H}$. Then $ p =\Pro_{K} x \Leftrightarrow \left[  p \in K, x-p \perp p,  ~ \text{and}~ x-p \in K^{\ominus}\right]$.
\end{fact}

\begin{fact} {\rm  \cite[page~48]{SS2016I} } \label{fact:coBS}
	Let $C$ and $D$ be nonempty convex subsets of $\mathcal{H}$
	such that $C \neq \{0\}$ and $D \neq \{0\}$.  Then
	\begin{align*}
	c_{0} (C,D) = \max \Big\{ 0, \sup \big\{   \innp{x,y} ~\big|~ x \in \overline{\cone} (C) \cap \mathbf{S}_{\mathcal{H}}, ~ y \in \overline{\cone} (D) \cap \mathbf{S}_{\mathcal{H}}\big\}   \Big\},
	\end{align*}
	where $\mathbf{S}_{\mathcal{H}}$ denotes the unit sphere in $\mathcal{H}$.
\end{fact}

\begin{fact} {\rm \cite[Lemma~2.4, Theorem~2.5 and Proposition~3.3]{DH2006II}}
	\label{fact:AnglesProperties}
		Let $C$ and $D$ be   nonempty convex subsets of $\mathcal{H}$.
		Then the following hold:
		\begin{enumerate}
			\item \label{fact:AnglesProperties:between0and1}$ c_{0} (C,D)  \in \left[0,1\right]$ and $ c (C,D)  \in \left[0,1\right]$.
			\item \label{fact:AnglesProperties:ineq} $(\forall x \in \overline{\cone} (C) )$ $(\forall y \in \overline{\cone} (D) )$ $\innp{x,y} \leq   c_{0} (C,D) \norm{x} \norm{y}$.
			\item  \label{fact:AnglesProperties:c0:EQ} $ c_{0} (C,D)=c_{0} (D, C)=c_{0} \left( \overline{ C} ,\overline{D} \right)=c_{0} \left(  \cone  (C) , \cone (D) \right)=c_{0} \left( \overline{\cone} (C) ,\overline{\cone} (D) \right)$.
		\end{enumerate}
	\end{fact}

\begin{fact} {\rm \cite[Lemma~2.11]{BOWJMAA}}\label{lemma:CD}
	Let $C$ and $D$ be   nonempty convex subsets of $\mathcal{H}$. Then the following hold:
	\begin{enumerate}
		\item \label{lemma:CD:UV} If $U$ and $V$ are nonempty convex subsets of
		$\mathcal{H}$ such that $C \subseteq U$ and $D \subseteq V$, then $c_{0}(C,D) \leq c_{0}(U,V)$.
		\item \label{lemma:CD:--}  $c_{0}(C,D) = c_{0}(-C,-D)$, $c_{0}(-C,D) = c_{0}(C,-D)$, $c (C,D) = c (-C,-D)$, and  $c (-C,D) = c (C,-D)$.

		\item  \label{lemma:CD:C0=1}
		If $ (\overline{\cone} (C) \cap \overline{\cone} (D) ) \smallsetminus \{0\} \neq \varnothing$, then $c_{0} (C,D)  =1$.

		\item \label{lemma:CD:c0:c} $0 \leq c(C,D) \leq c_{0} (C,D) \leq 1$.
	\end{enumerate}
\end{fact}

 \begin{fact} {\rm \cite[Theorem~4.6]{D2012}} \label{fact:K1:Km:dualsum}
Let $K_{1}$ and $K_2$ be nonempty closed convex cones in $\mathcal{H}$.
Then
\begin{equation*}
(K_1\cap K_2)^\ominus = \overline{K^{\ominus}_{1} + K_2^\ominus}.
\end{equation*}
 \end{fact}

\begin{fact} {\rm \cite[Propositions~3.3(4)]{DH2006II}} \label{fact:coneK1K2}
	Let  $K_{1}$ and $K_{2}$ be  nonempty  closed convex cones in $\mathcal{H}$.
	Then
	\begin{equation*}
		c(K_{1},K_{2}) = c_{0} \left(  K_{1}  \cap (K_{1}\cap K_{2})^{\ominus}, K_{2} \cap (K_{1}\cap K_{2})^{\ominus} \right).
	\end{equation*}
\end{fact}

 \begin{fact} {\rm \cite[Lemma~4.1]{BOWJMAA}}\label{lemma:cK1K2}
 	Let $K_{1}$ and $K_{2}$ be nonempty closed convex cones in $\mathcal{H}$. Then the following hold:
 	\begin{enumerate}
 		\item \label{lemma:cK1K2:1} If $K_{1} \cap K_{2} \neq \{0\}$, then $c_{0}(K_{1},K_{2}) =1$.
 		\item \label{lemma:cK1K2:0EQ} If $K_{1} \cap K_{2} =\{0\}$,  then $c_{0}(K_{1},K_{2}) =c(K_{1},K_{2}) $.

 		\item \label{lemma:cK1K2:H}  $K_{1} \cap K_{2} =\{0\}$ if and only if $ \overline{K_{1}^{\ominus}+K_{2}^{\ominus}} =\mathcal{H}$.

 	\end{enumerate}
 \end{fact}

\begin{fact} {\rm \cite[Proposition~4.4]{BOWJMAA}}\label{prop:cK1K2}
 Let $\mathcal{K}$ be a finite-dimensional linear subspace of $\mathcal{H}$, and
 let $K_{1}$ and $K_{2}$ be nonempty closed convex cones in $\mathcal{H}$ such that
 $K_{1} \subseteq  \mathcal{K}$ or $K_{2} \subseteq  \mathcal{K}$.
 Then the following hold:
	\begin{enumerate}
		\item \label{prop:cK1K2:c01} $K_{1} \cap K_{2} \neq \{0\}$ if and only if  $c_{0}(K_{1},K_{2}) =1$.
		\item \label{prop:cK1K2:EQ}   $K_{1} \cap K_{2} =\{0\}$ if and only if  $c_{0}(K_{1},K_{2}) =c(K_{1},K_{2}) $.
		\item \label{prop:cK1K2:c01:EQ} $K_{1} \cap K_{2} = \{0\}$ if and only if  $c_{0}(K_{1},K_{2}) <1$.
	\end{enumerate}
\end{fact}

\begin{fact} {\rm \cite[Theorem~4.7]{BOWJMAA}}\label{theorem:K1K2closed}
Let $K_{1}$ and $K_{2}$ be  nonempty closed convex cones in
$\mathcal{H}$. If $c_{0}(K_{1}, K_{2}) <1$,
then $K_{1} - K_{2}$ is closed.
\end{fact}

\begin{fact} {\rm \cite[Theorem~4.11]{BOWJMAA}}\label{theorem:KominusMperpNeq0}
	Let $K_{1} $ and $K_{2}$  be  closed convex cones in $\mathcal{H}$
	such that $K_{1} \cap K_{2}=\{0\}$ and $K_1$ is not linear.
	Furthermore, suppose that one of the following holds:
	\begin{enumerate}
		\item \label{theorem:KominusMperpNeq0:M} There exists $u \in \mathcal{H}$ such that $K_{2} =\{u  \}^{\perp}$.
		\item \label{theorem:KominusMperpNeq0:H} There exists $u \in \mathcal{H}$
		such that  $K_{2} \subseteq \{u\}^\perp$ and $\{u\}^\perp \cap K_{1} =\{0\}$.
		\item \label{theorem:KominusMperpNeq0:K}
		There exists a finite-dimensional linear subspace $\mathcal{K}$ of $\mathcal{H}$
		such that $K_{1} \subseteq  \mathcal{K}$ or $K_{2} \subseteq  \mathcal{K}$.
	\end{enumerate}
	Then
	\begin{align*}
	K^{\ominus}_{1} \cap K^{\oplus}_{2} \neq \{0\} \quad \text{and} \quad K^{\oplus}_{1} \cap K^{\ominus}_{2} \neq \{0\}.
	\end{align*}
\end{fact}

\section{Positive results}

\label{sec:pos}

We begin with a simple generalization of
\cite[Lemma~2.10(2)]{Deutsch1995} from two linear subspaces to one cone and one
linear subspace.

\begin{lemma} \label{lemma:MK}
Let $K  $ be a nonempty closed convex cone in $\mathcal{H} $ and let $M   $ be a
closed linear subspace of $\mathcal{H}$. Then
\begin{align*}
(\forall x \in K) (\forall  y \in M) \quad  \abs{\innp{x,y}} \leq c_{0}(K,M) \norm{x}\norm{y}.
\end{align*}
\end{lemma}
\begin{proof}
Let $x \in K$ and $ y \in M$. Then $-y\in M$ as well because $M$ is linear.
Applying
\cref{fact:AnglesProperties}\cref{fact:AnglesProperties:ineq} with $C=M$ and
$D=K$ to obtain that
$\pm\innp{x,y}=	\innp{x,\pm y} \leq c_{0}(K,M) \norm{x}\norm{\pm y}
=c_{0}(K,M) \norm{x}\norm{y}$
and the result follows.
\end{proof}

When one cone is contained in another, then the following pleasing result holds:

\begin{proposition} \label{prop:K1K1c=0}
Let $K_{1}$ and $K_{2}$ be nonempty closed convex cones in $\mathcal{H}$
such that $K_{1} \subseteq K_{2}$.
Then
\begin{equation*}
c(K_{1}, K_{2}) =0=c(K^{\ominus}_{1}, K^{\ominus}_{2}).
\end{equation*}
\end{proposition}
\begin{proof}
Because $K_{1} \subseteq K_{2}$, we have $K_{1} \cap K_{2}=K_{1}$ and $(K_{1} \cap
K_{2})^{\ominus}=K_{1}^{\ominus}$. Now $K_{1}$ is a nonempty closed
cone and so $0 \in K_{1}$. Hence, by \cref{lemma:K}, we have that $K_{2}=
\{0\} +K_{2} \subseteq K_{1}+K_{2} \subseteq K_{2} +K_{2} =K_{2}$,
which implies that $K_{1}+K_{2}= K_{2}$.
Combine this with \cref{fact:K1:Km:dualsum} and  \cref{fact:dualcone}\cref{fact:dualcone:closedconvex} to see that $(K_{1}^{\ominus} \cap K_{2}^{\ominus})^{\ominus} =\overline{K_{1}^{\ominus\ominus} +K_{2}^{\ominus\ominus} } =\overline{K_{1} +K_{2}}=\overline{K_{2}}=K_{2}$.
Moreover, using \cref{fact:coneK1K2} and \cref{defn:Angles}, we obtain
\begin{align*}
c(K_{1},K_{2}) &= c_{0} \left(  K_{1}  \cap (K_{1}\cap K_{2})^{\ominus}, K_{2} \cap (K_{1}\cap K_{2})^{\ominus} \right) = c_{0} \left(  K_{1}  \cap  K_{1}^{\ominus}, K_{2} \cap  K_{1}^{\ominus} \right)\\
&= c_{0} \left(  \{0\}, K_{2} \cap K_{1}^{\ominus} \right)=0
\end{align*}
and
\begin{align*}
c(K_{1}^{\ominus},K_{2}^{\ominus}) &= c_{0} \left(  K_{1}^{\ominus}  \cap (K_{1}^{\ominus} \cap K_{2}^{\ominus})^{\ominus}, K_{2}^{\ominus} \cap (K_{1}^{\ominus} \cap K_{2}^{\ominus})^{\ominus} \right)
= c_{0} \left(  K_{1}^{\ominus} \cap K_{2}, \{0\}\right)=0,
\end{align*}
which completes the proof.
\end{proof}

We are now ready for our first main result --- the concial extension of
Hundal's Lemma (\cref{f:Hundal}):

\begin{theorem} {\bf (concial extension of Hundal's Lemma)} \label{prop:KMoplus}
	Let $ K_{1}$ and $ K_{2}$  be nonempty closed convex cones in $ \mathcal{H}$.
	Suppose that $c_{0} (K_{1},K_{2}) <1$ and that $X$ is a convex subset of $\mathcal{H}$ which contains $K_{1}-K_{2}$.  Then
	\begin{align*}
	c_{0} (K_{1}, K_{2})  \leq c_{0} (K_{1}^{\oplus} \cap X , K_{2}^{\ominus} \cap X ) \leq c_{0} (K_{1}^{\oplus}   , K_{2}^{\ominus} ).
	\end{align*}
\end{theorem}

\begin{proof}
\cref{lemma:CD}\cref{lemma:CD:UV} yields
\begin{align*}
c_{0} (K_{1}^{\oplus} \cap (K_{1} -K_{2} ) , K_{2}^{\ominus} \cap (K_{1}-K_{2} ) )\leq c_{0} (K_{1}^{\oplus} \cap X , K_{2}^{\ominus} \cap X ) \leq c_{0} (K_{1}^{\oplus}, K_{2}^{\ominus} ).
\end{align*}
Hence, it suffices to prove that $c_{0} (K_{1}, K_{2})  \leq c_{0} (K_{1}^{\oplus} \cap (K_{1} -K_{2} ) , K_{2}^{\ominus} \cap (K_{1}-K_{2} ) )$.

If $c_{0} (K_{1},K_{2}) =0$, then $c_{0} (K_{1}, K_{2})  \leq c_{0} (K_{1}^{\oplus} \cap (K_{1} -K_{2} ) , K_{2}^{\ominus} \cap (K_{1}-K_{2} ) )$ is trivial.

Now assume that $c_{0} (K_{1},K_{2}) \in \left]0,1\right[\,$.
Then, by \cref{fact:coBS}, there exist sequences $(x_{k})_{k \in \mathbb{N}}$ in $K_{1} \cap \mathbf{S}_{\mathcal{H}}$ and  $(y_{k})_{k \in \mathbb{N}}$ in $K_{2} \cap \mathbf{S}_{\mathcal{H}}$ such that $\innp{x_{k}, y_{k}} \to c_{0} (K_{1},K_{2})$.

Clearly, $K_{1}^{\oplus}=-K_{1}^{\ominus} $.
Moreover, by \cref{fact:chara:PK},  $(\forall k \in \mathbb{N})$ $x_{k} -
\Pro_{K_{2}}x_{k} \in K^{\ominus}_{2} \cap (K_{1} -K_{2}) $ and
$\Pro_{K_{1}}\Pro_{K_{2}}x_{k}-\Pro_{K_{2}}x_{k} \in K_{1}^{\oplus} \cap (K_{1}
-K_{2})$.
Set
$c_{0}:=c_{0} (K_{1}, K_{2})$ and
$(\forall k \in \mathbb{N})$ $
\alpha_{k}:=\norm{\Pro_{K_{2}}x_{k} } $.
Hence, using \cref{lemma:CD}\cref{lemma:CD:UV}, \cref{fact:chara:PK},
and the same techniques employed
in the proof of \cite[Lemma~2.14]{Deutsch1995}, we obtain that $\alpha_{k} \to c_{0}$
and that for every $k \in \mathbb{N}$,
\begin{align*}
c_{0} (K_{1}^{\oplus} \cap (K_{1} -K_{2} ) , K_{2}^{\ominus} \cap (K_{1}-K_{2} ) )
&\geq   \frac{\biginnp{ \Pro_{K_{1}}\Pro_{K_{2}}x_{k}-\Pro_{K_{2}}x_{k} ,   x_{k} - \Pro_{K_{2}}x_{k} } }{\norm{\Pro_{K_{1}}\Pro_{K_{2}}x_{k}-\Pro_{K_{2}}x_{k}}  \norm{x_{k} - \Pro_{K_{2}}x_{k}}}\\
&\geq   \alpha_{k}- \sqrt{ \frac{c^{2}_{0} -\alpha^{2}_{k} }{ 1-c^{2}_{0}}} \to c_{0},
\end{align*}
which completes the proof.
\end{proof}

\begin{remark}
If $K_1$ and $K_2$ are linear in \cref{prop:KMoplus},
then $K_1-K_2=K_1+K_2$, $K_1^\oplus = K_1^\perp$,
$K_2^\ominus = K_2^\perp$, and we recover \cref{f:Hundal}.
\end{remark}

We conclude this section with further comments on \cref{{prop:KMoplus}} which are based
on the following example.

\begin{example}\label{exam:K1K2} {\rm \cite[Example~4.13]{BOWJMAA}}
	Suppose that $\mathcal{H} =\mathbb{R}^{2}$.
	Set $K_{1}:=\mathbb{R}^{2}_{+}$ and $K_{2}:= \{ (x_{1}, x_{2}) \in \mathbb{R}^{2} ~|~ -x_{1} \geq x_{2}   \}$.
	Then the following hold:
	\begin{enumerate}
		\item \label{exam:K1K2:ominus} $K_{1}^{\ominus}= \mathbb{R}^{2}_{-}$, $K_{2}^{\ominus}=\mathbb{R}_{+} (1,1)$, and $K_{2}^{\oplus} =\mathbb{R}_{+}(-1,-1)$.
		\item  \label{exam:K1K2:cap} $K_{1} \cap K_{2} =\{0\}$,  $K_{1}^{\ominus} \cap K_{2}^{\ominus} =\{0\}$, $K_{1}^{\ominus} \cap K_{2}^{\oplus} =K_{2}^{\oplus}$, $K_{1} +K_{2} =\mathbb{R}^{2}$, and $K_{1} - K_{2} =- K_{2}\neq\mathcal{H}$.
		\item \label{exam:K1K2:ineq} $c  (K_{1}, K_{2}) =c_{0} (K_{1}, K_{2}) =\tfrac{1}{\sqrt{2}} > 0=c_{0}  (K_{1}^{\ominus}, K_{2}^{\ominus}) =c_{0}  (K_{1}^{\oplus}, K_{2}^{\oplus}) =c (K_{1}^{\ominus}, K_{2}^{\ominus}) =c  (K_{1}^{\oplus}, K_{2}^{\oplus})$.

		\item \label{exam:K1K2:ineqoplus} $c_{0} (K_{1}^{\ominus},K_{2}^{\oplus}) =1$, $c  (K_{1}^{\ominus},K_{2}^{\oplus}) =0$, $c_{0} (K_{1},K_{2}) <  c_{0} (K_{1}^{\ominus},K_{2}^{\oplus}) $, and  $c  (K_{1},K_{2}) > c  (K_{1}^{\ominus},K_{2}^{\oplus}) $.
	\end{enumerate}
\end{example}

\begin{remark} {\rm \bf (importance of using dual-polar pair)}
Let $K_1$ and $K_2$ be as in \cref{exam:K1K2}.
Then $c_{0} (K_{1}, K_{2}) <1$ yet
\begin{equation*}
c_0(K_1,K_2) = \tfrac{1}{\sqrt{2}}
>
0 = c_0(K_1^\ominus,K_2^\ominus) = c_0(K_1^\oplus,K_1^\oplus).
\end{equation*}
This illustrates that in the conical extension of Hundal's Lemma
(\cref{prop:KMoplus}), we indeed must work with a pair of
dual-polar cones of the original pair --- substituting with
a polar-polar or dual-dual pair will not work!
\end{remark}

\section{Negative results}

\label{sec:neg}
We start with characterizations for differences of convex cones to be linear subspaces.

\begin{lemma}\label{l:difference}
Let $K_{1}$ and $K_{2}$ be nonempty closed convex cones in $\mathcal{H}$ such that
$K_{1}\cap K_{2}=\{0\}$. Then the following are equivalent:
\begin{enumerate}
\item\label{i:subspace} $K_{1}-K_{2}$ is a linear subspace of $\mathcal{H}$.
\item\label{i:superset} $(-K_1)\cup K_2 \subseteq K_{1}-K_{2}$.
\item\label{i:bothsub} $K_{1}$ and $K_{2}$ are linear subspaces of $\mathcal{H}$.
\end{enumerate}
\end{lemma}

\begin{proof}
 ``\cref{i:subspace}$\Rightarrow$\cref{i:superset}'':
 We have
 $K_1 = K_1 - \{0\} \subseteq K_1-K_2$ and
 $-K_2 = \{0\}-K_2 \subseteq K_1-K_2$.
 Hence
 $(-K_1)\cup K_2 = -(K_1\cup (-K_2)) \subseteq -(K_1-K_2)=K_1-K_2$.

 ``\cref{i:superset}$\Rightarrow$\cref{i:bothsub}'':
 Let $x\in -K_{1}$. Then there exist $y_1\in K_1$ and $y_2\in K_2$ such that
 $x=y_1-y_2$.
 Then $y_2 = y_1-x \in K_1 - (-K_1) = K_1 + K_1 = K_1$.
 Hence $y_2 \in K_1\cap K_2 = \{0\}$.
 Thus $y_2 = 0$ and so $x = y_1\in K_1$.
 This implies $-K_1\subseteq K_1$ and therefore
 $K_1$ is linear by \cref{lemma:K}.
 The argument for $K_2$ is similar.

 ``\cref{i:bothsub}$\Rightarrow$\cref{i:subspace}'': Obvious.
\end{proof}

Here is a conical variant of \cref{f:KKM}.

\begin{theorem} \label{theorem:K1K2ominusoplusEQ}
Let $ K_{1}$ and $ K_{2}$  be  nonempty closed convex cones in $ \mathcal{H}$
such that $K_{1} \cap K_{2} =\{0\}$ and $K_1-K_2=\mathcal{H}$.
Then
\begin{enumerate}
\item  \label{theorem:K1K2ominusoplusEQ:linear} $K_{1}$ and $K_{2}$ are linear subspaces
of $\mathcal{H}$.
\item  \label{theorem:K1K2ominusoplusEQ:eq}
$c_{0} (K_{1}, K_{2})  =  c_{0} (K_{1}^{\oplus}   , K_{2}^{\ominus}   )= c_{0} (K_{1}^{\ominus}   , K_{2}^{\oplus}   )=
c_{0}(K_{1}^{\perp}, K_{2}^{\perp}).$
\end{enumerate}
\end{theorem}

\begin{proof}
\cref{theorem:K1K2ominusoplusEQ:linear}:
Clear from \cref{l:difference}.
\cref{theorem:K1K2ominusoplusEQ:eq}:
In view of \cref{theorem:K1K2ominusoplusEQ:linear},
we have $-K_2=K_2$ and thus $K_1+K_2=K_1-K_2=\mathcal{H}$.
Now apply \cref{f:KKM}.
\end{proof}

\begin{remark}{\bf (impossibility of extending \cref{f:KKM} to cones)}
The assumption on the two subspaces in \cref{f:KKM} is that
their intersection is $\{0\}$ and their sum is the entire space.
Now assume that $K_2$ is a linear subspace
and $K_1+K_2=K_1-K_2$.
Then Lemma~\ref{l:difference} implies that $K_1$ is a subspace as well.
Hence \cref{f:KKM} does not appear to admit an extension to the case
when one linear subspace is replaced by a convex cone!
\end{remark}

\begin{remark}{\bf (importance of the assumption that $K_1-K_2=\mathcal{H}$)}
Let $K_1$ and $K_2$ be as in \cref{exam:K1K2}.
Then $K_1\cap K_2 = \{0\}$ but neither $K_1$ nor $K_2$ is a linear subspace.
This shows that the assumption
that $\mathcal{H} = K_{1}-K_{2}$  in \cref{theorem:K1K2ominusoplusEQ}
is critical.
\end{remark}

\begin{remark}{\bf (importance of the assumption that $K_1\cap K_2=\{0\}$)}
Let $u_{1}, u_{2}$ be unit vectors in $\mathbb{R}^3$ such that $0<\scal{u_{1}}{u_{2}}<1$.
Set $K_1:=\{u_{1}\}^{\perp}$ and $K_2=\{u_{2}\}^{\perp}$. Then
$K_{1}^{\perp}=\mathbb{R}u_{1}$,
$K_{2}^{\perp}=\mathbb{R}u_{2}$,
$K_1\cap K_2\neq \{0\}$ and $K_{1}-K_{2}=\mathbb{R}^{3}$, but
$c_{0}(K_{1}^{\perp},K_{2}^{\perp})=\scal{u_{1}}{u_{2}}<1=c_{0}(K_{1},K_{2})$.
\end{remark}

We now turn to a conical variant of \cref{f:Solmon}.
Our proof is an adaptation of Deutsch's proof of \cite[Theorem~2.16]{Deutsch1995}.

\begin{theorem} \label{theorem:cEQ}
	Let $K_{1}$ and $K_{2}$ be nonempty closed convex cones in $\mathcal{H}$.
	Suppose that $c(K_{1},K_{2}) <1$, that $K_{1} \cap K_{2}$  and
	$K_{1}^{\oplus} \cap K_{2}^{\ominus}$  are linear subspaces of
	$\mathcal{H}$, that $\overline{K_{1}^{\oplus} +(K_{1} \cap K_{2})} \cap
	(K_{1} \cap K_{2})^{\perp} =K_{1}^{\oplus}$, and that $\overline{K_{2}^{\ominus}
	+(K_{1} \cap K_{2})} \cap (K_{1} \cap K_{2})^{\perp} =K_{2}^{\ominus}$. Then
	\begin{align*}
	c(K_{1}, K_{2})  \leq  c(K_{1}^{\oplus}, K_{2}^{\ominus}).
	\end{align*}
	If additionally $c(K_{1}^{\oplus},K_{2}^{\ominus}) <1$,
	\begin{align*}
	\overline{K_{1} +(K_{1}^{\oplus} \cap K_{2}^{\ominus})}   \cap (K_{1}^{\oplus} \cap K_{2}^{\ominus})^{\perp}=K_{1}, \quad \text{and} \quad \overline{K_{2} +(K_{1}^{\oplus} \cap K_{2}^{\ominus})} \cap (K_{1}^{\oplus} \cap K_{2}^{\ominus})^{\perp}=K_{2},
	\end{align*}
	then
	\begin{align*}
	c(K_{1}, K_{2})  =  c(K_{1}^{\oplus}, K_{2}^{\ominus}).
	\end{align*}
\end{theorem}

\begin{proof}
	\cref{fact:coneK1K2} and \cref{fact:dualcone}\cref{fact:dualcone:o} imply that
	\begin{align}\label{eq:prop:cEQ:leq:c0}
	c_{0} \big(  K_{1}  \cap (K_{1}\cap K_{2})^{\perp}, K_{2} \cap (K_{1}\cap K_{2})^{\perp} \big) =c(K_{1},K_{2}) <1.
	\end{align}
	Because $K_{1}^{\oplus} \cap   K_{2}^{\ominus}$ is a linear subspace of
	$\mathcal{H}$, \cref{fact:K1:Km:dualsum},
	\cref{lemma:polardual}\cref{lemma:polardual:llinear} and
	\cref{fact:dualcone}\cref{fact:dualcone:o}\&\cref{fact:dualcone:linearsubsp} imply
	that  $(K_{1}^{\oplus} \cap   K_{2}^{\ominus})^{\perp} =\overline{
	K_{1}-  K_{2} }$ is a linear subspace of $\mathcal{H}$. Hence
	\begin{align} \label{eq:prop:cEQ:X}
	X:=  (K_{1} \cap K_{2})^{\perp} \cap (K_{1}^{\oplus} \cap   K_{2}^{\ominus})^{\perp} =   (K_{1} \cap K_{2})^{\perp} \cap  \overline{ K_{1}-  K_{2} }
	\end{align}
	is a closed linear subspace of $\mathcal{H}$.
	Because $\overline{ K_{1}-  K_{2} }$ is a closed linear subspace of $\mathcal{H}$,
	we learn that $K_{2}   \subseteq \overline{ K_{1}-  K_{2} }$
	and that $K_{1} \subseteq \overline{ K_{1}-  K_{2} }$.
	Hence
	\begin{align} \label{eq:prop:cEQ:KX}
	K_{1} \cap X =  K_{1} \cap (K_{1} \cap K_{2})^{\perp}
	\quad\text{and}\quad
	K_{2} \cap X =  K_{2} \cap (K_{1} \cap K_{2})^{\perp}.
	\end{align}
	Moreover,  $(K_{1} \cap K_{2})^{\perp}$ is a linear subspace and so $(K_{1}
	\cap K_{2})^{\perp} - (K_{1} \cap K_{2})^{\perp} = (K_{1} \cap
	K_{2})^{\perp}$.
	Combine this with \cref{eq:prop:cEQ:X} and
	\cref{eq:prop:cEQ:KX} to see that
	\begin{align} \label{eq:prop:cEQ:subseteqX}
	\big(K_{1} \cap (K_{1} \cap K_{2})^{\perp}\big)-
	\big(K_{2} \cap (K_{1} \cap K_{2})^{\perp}\big) =
	\big(K_{1} \cap X\big)  - \big(K_{2} \cap X\big) \subseteq X.
	\end{align}
	Using \cref{fact:K1:Km:dualsum},
	\cref{lemma:polardual}\cref{lemma:polardual:-C}$\&$\cref{lemma:polardual:oplus}
	and the assumption  that $\overline{K_{1}^{\oplus} +K_{1} \cap K_{2}} \cap (K_{1}
	\cap K_{2})^{\perp} =K_{1}^{\oplus}$,
	we obtain
\begin{subequations}
\label{eq:prop:cEQ:KXoplus}
\begin{align}
(K_{1} \cap (K_{1} \cap K_{2})^{\perp})^{\oplus} \cap X  &=  \overline{K_{1}^{\oplus} +K_{1} \cap K_{2}} \cap (K_{1} \cap K_{2})^{\perp} \cap (K_{1}^{\oplus} \cap   K_{2}^{\ominus})^{\perp}\\
&=K_{1}^{\oplus}  \cap (K_{1}^{\oplus} \cap   K_{2}^{\ominus})^{\perp}.
\end{align}
\end{subequations}
	Similarly, using \cref{fact:K1:Km:dualsum},
	\cref{lemma:polardual}\cref{lemma:polardual:-C},
	\cref{fact:dualcone}\cref{fact:dualcone:closedconvex} and the assumption
	that $\overline{K_{2}^{\ominus} +K_{1} \cap K_{2}} \cap (K_{1} \cap
	K_{2})^{\perp} =K_{2}^{\ominus}$,
	we see that
\begin{subequations}
\label{eq:prop:cEQ:MXperp}
\begin{align}
(K_{2} \cap (K_{1} \cap K_{2})^{\perp})^{\ominus} \cap X &=  \overline{K_{2}^{\ominus} +K_{1} \cap K_{2}} \cap (K_{1} \cap K_{2})^{\perp} \cap (K_{1}^{\oplus} \cap   K_{2}^{\ominus})^{\perp} \\
&=K_{2}^{\ominus}\cap(K_{1}^{\oplus} \cap   K_{2}^{\ominus})^{\perp}.
\end{align}
\end{subequations}
	We now use  \cref{eq:prop:cEQ:leq:c0}, \cref{eq:prop:cEQ:subseteqX}, and
	\cref{prop:KMoplus} (with $K_{1} = K_{1} \cap X$, $K_{2}=K_{2} \cap X$, and
	$X$ is as   \cref{eq:prop:cEQ:X}) to deduce that $c_{0} (  K_{1} \cap X ,
	K_{2} \cap X ) \leq c_{0} ((K_{1} \cap X)^{\oplus} \cap X,  (K_{2} \cap X
	)^{\ominus} \cap X)$, which, recalling \cref{eq:prop:cEQ:KX},
	\cref{eq:prop:cEQ:KXoplus}, and \cref{eq:prop:cEQ:MXperp}, is equivalent to
	\begin{align*}
	c_{0} (K_{1} \cap (K_{1} \cap K_{2})^{\perp} , K_{2} \cap (K_{1} \cap K_{2})^{\perp} ) \leq c_{0} \left(K_{1}^{\oplus}  \cap (K_{1}^{\oplus} \cap   K_{2}^{\ominus})^{\perp}, K_{2}^{\ominus}\cap(K_{1}^{\oplus} \cap   K_{2}^{\ominus})^{\perp}\right).
	\end{align*}
This is	--- using \cref{fact:coneK1K2} and \cref{fact:dualcone}\cref{fact:dualcone:o} ---
the same as $c(K_{1} ,K_{2} )  \leq c(K_{1}^{\oplus} , K_{2}^{\ominus})$.

	Finally, with the additional assumptions,
	apply the result just proved above  to
	$K_{1} =K_{1}^{\oplus}$ and $K_{2} =K_{2}^{\ominus}$ to conclude that
	$c(K_{1}^{\oplus} , K_{2}^{\ominus}) \leq  c(K_{1}^{\oplus \oplus} , K_{2}^{\ominus \ominus})  = c(K_{1},K_{2}) $, where the last equation
	follows from \cref{fact:dualcone}\cref{fact:dualcone:closedconvex}  and \cref{lemma:polardual}\cref{lemma:polardual:oplus}.
	Altogether, $c(K_{1},K_{2})  = c(K_{1}^{\oplus} , K_{2}^{\ominus})$ in this case.
\end{proof}

To derive some consequences of \cref{theorem:cEQ}, we require the following result.

\begin{lemma} \label{lemma:KMperp}
Let $A$ and $B$ be nonempty subsets of  $\mathcal{H}$ such that
$0 \in B$ and $A \subseteq B^{\perp}$.
Then
\begin{equation*}
(A+B) \cap B^{\ominus} =A.
\end{equation*}
Consequently, if $B$ is a linear subspace of $\mathcal{H}$,
then $(A+B) \cap B^{\perp} =A$.
\end{lemma}
\begin{proof}
Because  $0 \in B$ and
$A \subseteq  B^{\ominus}$, we have that $ A \subseteq (A+B) \cap B^{\ominus}$.
Conversely, let $x \in  (A+B) \cap B^{\ominus}$. Then there exist $a \in A$ and
$b \in B$ such that  $x =a+b \in B^{\ominus}$.
Because $x \in B^{\ominus}$,
$b \in B$, and $a \in A \subseteq B^\perp = B^{\ominus} \cap B^{\oplus}$, we know that
$\innp{x, b} \leq 0$, and that $\innp{a, b} =0$.
Hence $0
\geq \innp{x,b}=\innp{a+b,b}=\innp{a,b}+\norm{b}^{2} =\norm{b}^{2} \geq 0$
so $\norm{b}^2 = 0$, i.e., $b=0$.
Thus $x=a\in A$ and so
$(A+B) \cap B^{\ominus} \subseteq A$.

The ``Consequently'' part follows from what we just proved and
\cref{lemma:polardual}\cref{lemma:polardual:llinear} which states that
$B^{\perp}=B^{\ominus}=B^{\oplus}$ provided that $B$ is linear.
\end{proof}

We first reprove Solmon's results (\cref{f:Solmon}) from \cref{theorem:cEQ}.

\begin{corollary}{\bf (Solmon)} \label{cor:cEQ}
Let  $K_{1}$ and $K_{2}$ be closed linear subspaces of $\mathcal{H}$. Then
$c(K_{1}, K_{2})  =  c(K_{1}^{\perp}, K_{2}^{\perp})$.
\end{corollary}
\begin{proof}
Because  $K_{1}$  and $K_{2}$ are closed linear subspaces,
by \cite[Theorem~2.13]{Deutsch1995},
\begin{align}\label{eq:cor:cEQ}
c(K_{1},K_{2})   =1 \Leftrightarrow   K_{1} +K_{2} \text{ is not closed} \Leftrightarrow K_{1}^{\perp} +K_{2}^{\perp} \text{ is not closed}  \Leftrightarrow c(K_{1}^{\perp},K_{2}^{\perp}) =1.
\end{align}
Assume that $c(K_{1},K_{2})   <1$.
By \cref{eq:cor:cEQ},
$c(K_{1}^{\perp},K_{2}^{\perp})   <1$.  Note that
$K_{1}^{\perp}=K_{1}^{\ominus}=K_{1}^{\oplus}$, and
$K_{2}^{\perp}=K_{2}^{\ominus}=K_{2}^{\oplus}$.  Moreover, because
$K_{1}^{\perp} \perp (K_{1} \cap K_{2})$, then by
\cite[Proposition~29.6]{BC2017}, $\overline{K_{1}^{\perp} +(K_{1} \cap K_{2})}
=K_{1}^{\perp} +K_{1} \cap K_{2}$. Hence, by \cref{lemma:KMperp},
$\overline{K_{1}^{\oplus} +K_{1} \cap K_{2}} \cap (K_{1} \cap K_{2})^{\ominus} =
(K_{1}^{\perp} +K_{1} \cap K_{2} )\cap (K_{1} \cap K_{2})^{\ominus}
=K_{1}^{\perp}=K_{1}^{\oplus}$. Similarly,
$\overline{K_{2}^{\ominus} -K_{1} \cap K_{2}} \cap (K_{1} \cap K_{2})^{\ominus}
=K_{2}^{\ominus}$.
Replacing $(K_{1},K_{2})$ with $(K_{1}^{\perp},K_{2}^{\perp})$ in the above,
we obtain $ \overline{K_{1} +(K_{1}^{\oplus} \cap K_{2}^{\ominus})}   \cap (K_{1}^{\oplus}
\cap K_{2}^{\ominus})^{\perp}=K_{1}$ and $ \overline{K_{2} -(K_{1}^{\oplus} \cap
K_{2}^{\ominus})} \cap (K_{1}^{\oplus} \cap K_{2}^{\ominus})^{\perp}=K_{2}$.
Hence, by \cref{theorem:cEQ}, $c(K_{1}, K_{2})  =  c(K_{1}^{\oplus},
K_{2}^{\ominus}) =c(K_{1}^{\perp}, K_{2}^{\perp})$.
\end{proof}

Next we give some special cases of \cref{theorem:cEQ}.
\begin{corollary} \label{cor:prop:cEQ}
Let $K_{1}$ and $K_{2}$ be nonempty closed convex cones in $\mathcal{H}$ such that
$c(K_{1},K_{2}) <1$, $K_{1} \cap K_{2}$  and
$K_{1}^{\oplus} \cap K_{2}^{\ominus}$  are linear subspaces of
$\mathcal{H}$, and $ K_{1}^{\oplus} +(K_{1} \cap K_{2}) $  and $
K_{2}^{\ominus} +(K_{1} \cap K_{2}) $ are closed. Then
\begin{equation*}
c(K_{1}, K_{2})  \leq  c(K_{1}^{\oplus}, K_{2}^{\ominus}).
\end{equation*}
If $c(K_{1}^{\oplus}, K_{2}^{\ominus}) <1$,  and $K_{1}
	+(K_{1}^{\oplus} \cap K_{2}^{\ominus})$  and $K_{2} +(K_{1}^{\oplus} \cap
	K_{2}^{\ominus}) $ are closed, then
\begin{equation*}
c(K_{1}, K_{2})  =  c(K_{1}^{\oplus}, K_{2}^{\ominus}).
\end{equation*}
\end{corollary}
\begin{proof}
Because $K_{1} \cap K_{2}$  and $K_{1}^{\oplus} \cap K_{2}^{\ominus}$  are linear subspaces of $\mathcal{H}$, by \cref{fact:dualcone}\cref{fact:dualcone:linearsubsp}, we have that $ K_{1}^{\oplus} \subseteq (K_{1} \cap K_{2})^{\perp} $, $ K_{2}^{\ominus} \subseteq (K_{1} \cap K_{2})^{\perp} $, $K_{1} \subseteq (K_{1}^{\oplus} \cap K_{2}^{\ominus})^{\perp}$  and $K_{2} \subseteq (K_{1}^{\oplus} \cap K_{2}^{\ominus})^{\perp}$.

Hence, applying \cref{lemma:KMperp} with $A =K_{1}^{\oplus}$ and $B= K_{1} \cap
K_{2}$, with  $A =K_{2}^{\ominus}$ and $B= K_{1} \cap K_{2}$, with $A=K_{1}$ and
$ B= K_{1}^{\oplus} \cap K_{2}^{\ominus}$, and with $A=K_{2}$ and $ B=
K_{1}^{\oplus} \cap K_{2}^{\ominus}$,  we obtain that  $(K_{1}^{\oplus} +(K_{1}
\cap K_{2})) \cap (K_{1} \cap K_{2})^{\perp} =K_{1}^{\oplus}$,
$(K_{2}^{\ominus}
+(K_{1} \cap K_{2})) \cap (K_{1} \cap K_{2})^{\perp} =K_{2}^{\ominus}$,
$(K_{1} +(K_{1}^{\oplus} \cap K_{2}^{\ominus}))   \cap (K_{1}^{\oplus} \cap
K_{2}^{\ominus})^{\perp}=K_{1}$, and $(K_{2} +(K_{1}^{\oplus} \cap
K_{2}^{\ominus})) \cap (K_{1}^{\oplus} \cap K_{2}^{\ominus})^{\perp}=K_{2}$,
respectively.
Therefore, the required results follow from the closedness assumptions and  \cref{theorem:cEQ}.
\end{proof}

\begin{corollary}  \label{propo:cKM:Rn}
Let $K_{1}$ and $K_{2}$ be  nonempty closed convex cones in $\mathcal{H}$.
Suppose that $K_{1}$ or $K_{2}$ is contained in a finite-dimensional
linear subspace of $ \mathcal{H} $, and that  $K_{1}^{\oplus} $  or
$K_{2}^{\ominus} $ is  contained in a finite-dimensional linear subspace of $
\mathcal{H} $. Suppose furthermore that    $K_{1} \cap K_{2} =\{0\}$ and $K_{1}^{\oplus}
\cap K_{2}^{\ominus}=\{0\}$.
Then
\begin{align*}
c_{0}(K_{1},K_{2}) =	c(K_{1},K_{2}) =
c(K_{1}^{\oplus},K_{2}^{\ominus})=c_{0}(K_{1}^{\oplus},K_{2}^{\ominus}).
\end{align*}
\end{corollary}
\begin{proof}
Because $K_{1} \cap K_{2} =\{0\}$, $K_{1}^{\oplus} \cap K_{2}^{\ominus}=\{0\}$,
using the assumptions that $K_{1}$ or $K_{2}$ is contained in a
finite-dimensional linear subspace of $ \mathcal{H} $, and that
$K_{1}^{\oplus} $  or $K_{2}^{\ominus} $ is  contained in a finite-dimensional
linear subspace of $ \mathcal{H} $,   by
\cref{prop:cK1K2}, we have that
\begin{align*}
c (K_{1},K_{2}) =	c_{0}(K_{1},K_{2}) <1 \quad \text{and} \quad   c (K_{1}^{\oplus},K_{2}^{\ominus}) = c_{0}(K_{1}^{\oplus},K_{2}^{\ominus})<1.
\end{align*}
Hence, the desired identities are directly from \cref{cor:prop:cEQ}.
\end{proof}

Here is a simple illustration of  \cref{propo:cKM:Rn}.

\begin{example}
\label{ex:210508a}
Suppose that
$\mathcal{H} =\mathbb{R}^{2}$, and
set $K_{1} :=\mathbb{R}(1,0)$ and $K_{2} :=\mathbb{R}(1,1)$.
Then
\begin{equation*}
c_{0}(K_{1},K_{2}) =	c(K_{1},K_{2}) =
c(K_{1}^{\oplus},K_{2}^{\ominus})=c_{0}(K_{1}^{\oplus},K_{2}^{\ominus})
=\tfrac{1}{\sqrt{2}}.
\end{equation*}
\end{example}

The next example follows easily from the definitions and \cref{lemma:CD}\cref{lemma:CD:--}.
We will then comment on its relevance to previous results.

\begin{example} \label{example:KM}
Suppose that $\mathcal{H} =\mathbb{R}^{2}$,
and set  $K:=\{ (x_{1},x_{2}) \in \mathbb{R}^{2} ~|~ x_{2} \geq x_{1} \geq 0  \}$
and $M:=\mathbb{R}(1,0)$.
Then
 $K\cap M =\{0\}$, $K^{\ominus} =\{ (x_{1},x_{2}) \in \mathbb{R}^{2} ~|~
-x_{1} \geq x_{2} \text{ and } x_{2} \leq 0 \}$, $M^{\perp}=\mathbb{R}(0,1)$,
$K^{\ominus} \cap M^{\perp}= \mathbb{R}_{+}(0,-1)$,
 $(K^{\ominus} \cap M^{\perp} )^{\ominus} = \mathbb{R}\times\mathbb{R}_+$, and
 \begin{align*}
0&= c(K^{\oplus},M^{\perp})= c(K^{\ominus},M^{\perp}) \\
&<\tfrac{1}{\sqrt{2}} = c_{0}(K,M)=c(K,M)\\
&<1 = c_{0}(K^{\oplus},M^{\perp})=c_{0}(K^{\ominus},M^{\perp}).
 \end{align*}
\end{example}

\begin{remark}{\bf (impossiblility of a simple extension of Solmon's Fact)}
\cref{example:KM} very clearly shows the impossibility of
a nice and simple generalization of \cref{f:Solmon} from
two subspaces to even just a subspace and a cone:
Indeed,
in  \cref{example:KM}, we have
\begin{equation*}
c  (K^{\ominus}, M^{\perp}) =c (K^{\oplus}, M^{\perp})
< c(K,M);
\end{equation*}
thus, neither the polar cone $K^\ominus$ nor the dual cone $K^\oplus$ will
do the job!
\end{remark}

\begin{remark} {\bf (importance of linearity of $K_1^\oplus \cap K_2^\ominus$)}
Let us revisit  \cref{theorem:cEQ} with
$K_{1}$ and $K_{2}$  replaced by the $M$ and $K$
from \cref{example:KM}, respectively.
Then
$c(K_1,K_2) = \tfrac{1}{\sqrt{2}} < 1$
and
$K_1\cap K_2 = \{0\}$.
The latter clearly implies
$\overline{K_{1}^{\oplus} +(K_{1} \cap K_{2})} \cap
(K_{1} \cap K_{2})^{\perp} = K_{1}^{\oplus}$
and
$\overline{K_{2}^{\ominus}
+(K_{1} \cap K_{2})} \cap (K_{1} \cap K_{2})^{\perp} =K_{2}^{\ominus}$.
However,
$K_1^\oplus \cap K_2^\ominus =  \mathbb{R}_+(0,-1)$ and so
\begin{equation*}
\text{$K_1^\oplus \cap K_2^\ominus$ is \emph{not} a linear subspace,
and $c(K_1,K_2)>c(K_1^\oplus,K_2^\ominus)$.}
\end{equation*}
This shows that the assumption that $K_1^\oplus \cap K_2^\ominus$ be a linear subspace
is critical in \cref{theorem:cEQ}.
The very same example can be used to show the importance of the
assumption that $K_1^\oplus \cap K_2^\ominus$ be a linear subspace
in \cref{cor:prop:cEQ}.
\end{remark}

\begin{remark}{\bf (assumptions in \cref{theorem:cEQ} are quite restrictive)}
\label{rem:theorem5.5R2}
Now suppose that $\mathcal{H} =\mathbb{R}^{2}$. Let us assume that $K_{1}$ and $K_{2}$
are two nonempty closed convex cones in $\mathbb{R}^{2}$ such that $K_{1}$
or $K_{2}$ is not a linear subspace. Furthermore, assume that $K_{1} \cap K_{2} $ is
a linear subspace.
By \cref{theorem:KominusMperpNeq0}\cref{theorem:KominusMperpNeq0:K}, the
intersections
$K_{1} \cap K_{2} $ and $K_{1}^{\oplus} \cap K_{2}^{\ominus}$ cannot be both
equal to $\{(0,0)\}\subseteq \mathbb{R}^2$.
Because $K_{1}^{\oplus \oplus} \cap K_{2}^{\ominus  \ominus}
= K_{1} \cap K_{2}$, we assume without loss of generality that  $K_{1} \cap
K_{2} =\{0\}\times \mathbb{R}$.
Without loss of generality, we can only have one of the following
two cases:
\emph{Case~1}: $K_{1}=\mathbb{R}_+\times\mathbb{R}$ and $K_{2}=\mathbb{R}_-\times\mathbb{R}$;
\emph{Case~2}:
$K_{1}= \{0\}\times\mathbb{R}$ and  and $K_{2}=\mathbb{R}_-\times\mathbb{R}$.
But in either case, we have
$K_{1}^{\oplus} \cap K_{2}^{\ominus} =\mathbb{R}_{+}\times\{0\}$
which is \emph{not} a linear subspace.

Therefore, in the Euclidean plane $\mathbb{R}^{2}$, there do not exist two nonlinear cones satisfying
the assumptions in \cref{theorem:cEQ}!
\end{remark}

\cref{rem:theorem5.5R2} and \cref{l:difference} now prompt
the following natural question with which we conclude this paper:

\begin{question}
Do there exist nonempty closed convex cones $K_1$ and $K_2$
that are \emph{nonlinear} yet satisfy
the assumptions in \cref{theorem:cEQ}?
\end{question}

\section*{Acknowledgements}
HHB and XW were partially supported by NSERC Discovery Grants.

\addcontentsline{toc}{section}{References}

\bibliographystyle{abbrv}


text:
St. Peters Blue Angels and Delphic Newtown United ladies to do battle for first ever female FA Cup title.
BASSETERRE, St. Kitts,15th February, 2013- The SKNFA's Female FA Cup will come to an end on Sunday with St. Peters Blue Angels and Delphic Newtown United after last night's results. In the first match of the evening, St.Peters Blue Angels defeated St. Thomas Trinity Strikers 4 goals to 3. At the end of regular time neither teams had managed to score which meant that the winner would have to be decided by penalty kicks. At the end of five kicks, the game was still tied, now at 2 goals each. The game then went down to "sudden death" where St. Peters Blue Angels eventually emerged victorious.
In the second game of the evening Delphic Newtown United continued on in their winning ways when they defeated Conaree Lady Spartans three goals to nil in regular time.
The finals of the SKNFA's Female FA Cup is scheduled for the Verchilds Playing Field at 4:00 p.m.
MAY THE BEST TEAM WIN.
PressUncategorized
SKNFA Female FA Cup Semi-finals
Press Release: SKNFA Male FA Cup to begin this month.

redpajama_set_name: RedPajamaCommonCrawl


Winter is starting to become a reality. We just had a cord of FIREWOOD delivered this morning for the winter. Kind of exciting but not quite ready yet for Mother Natures fury. My first winter in WY was a mild one and this year is suppose to be much worse. You just never know what you are going to get. I do know that I will never have a house without a wood fireplace. They are so warm and comforting.
Now back to the yummy food.
I am so excited about this recipe. Doesn't it look delicious? I love every ingredient in this and so does my family. Anytime I can find a family favorite I am super happy. PLUS, my husband can make this with no problem which helps a lot while I am working.
I found this recipe on Six Sisters Stuff You have probably seen there blog before but if you haven't you are in for a treat. They have some wonderful recipes. Darling blog and I am a frequent follower of them. Love it.


text:
SLTN View
No end to wholesale woes
OCTOBER 8, 2012 will go down as a dark day in the Scottish licensed trade's history.
Administrators acting on behalf of WaverleyTBS announced that there would be no recovery for the stricken wholesaler. It was a sudden and dramatic end for a distinguished and long-established company, which had only entered administration on October 2.
Of course, no sooner had the shock set in than Waverley's rivals were moving quickly to profit from its demise – such is the nature of business.
On that basis, the disruption caused to Waverley customers will hopefully have been kept to a minimum: with so many alternative sources of supply eager to take its place, no one should have had stocking issues for long.
That there is plenty of choice for operators when it comes to the wholesale market is likely to have been one of the reasons why Waverley went down in the first place.
Not everyone in the drinks wholesale industry believes that there is a situation of over capacity in the market (perhaps some just don't want to admit it). But there seems little doubt that companies have been battling for a share of a diminishing market in recent years. The amount of pubs and other on-trade businesses that have closed since the economic downturn began tells you that.
The flip side of the coin is that the intense competition in the drinks supply business has, in some ways, been good for the on-trade.
As wholesalers have strived to retain customers, they have upped the service they provide in various ways. There has been talk of wholesalers offering more flexible credit terms, and in the case of the wine suppliers, a notable increase in support for outlets in terms of wine list development and staff training. There has also been more concerted moves into online ordering, giving greater flexibility to operators in buying terms.
If anything, with no immediate signs of economic recovery, the wholesale companies are going to have to find even more ways to support the trade.
Not only is this good for pubs, hotels and restaurants (and consumers in return), but for the wholesale sector itself.
Unfortunately, this will do nothing to help the many hundreds of staff (685 at the time of writing) who have already been made redundant by Waverley.
That amounts to an awful lot of families up and down the UK who have lost a salary, who will be very worried about where a replacement is going to come from in these austere times. Many will be highly skilled and more than capable of doing similar jobs for other companies in the sector, but given the intense competition that exists in the drinks wholesale market it's hard to see how many will remain in the sector. Indeed, if predictions of further consolidation are to come true, the job market is only going to get tighter in the world of drinks wholesaling.
The end of Waverley might bring a short-term boost to some of its rivals, but I suspect the problems have only just begun for the wholesale market.


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text:
Q: How can I reduce the time taken to extract files? I made a program in C# where it processes about 30 zipped folders which have about 35000 files in total. My purpose is to read every single file for processing its information. As of now, my code extracts all the folders and then read the files. The problem with this process is it takes about 15-20 minutes for it to happen, which is a lot.
I am using the following code to extract files:
void ExtractFile(string zipfile, string path)
{
    ZipFile zip = ZipFile.Read(zipfile);
    zip.ExtractAll(path);
}

The extraction part is the one which takes the most time to process. I need to reduce this time. Is there a way I can read the contents of the files inside the zipped folder without extracting them? or if anyone knows any other way that can help me reduce the time of this code ?
Thanks in advance

A: Maybe instead of extracting it to the hard disk, you should try read it without extraction, using OpenRead, then you would have to use the ZipArchiveEntry.Open method.
Also have a look at the CodeFluent Runtime tool, which claims to be improved for performances issues.

A: You could try reading each entry into a memory stream instead of to the file system:
ZipFile zip = ZipFile.Read(zipfile);
foreach(ZipEntry entry in zip.Entries)
{
    using(MemoryStream ms = new MemoryStream())
    {
        entry.Extract(ms);
        ms.Seek(0,SeekOrigin.Begin);
        // read from the stream
    }

}

A: Try to break your responses into single await async methods, which started one by one if one of the responses is longer than 50 ms. http://msdn.microsoft.com/en-us/library/hh191443.aspx
If we have for example 10 executions which call one by one, in async/await we call our executions parallel, and operation will depend only from server powers.

redpajama_set_name: RedPajamaStackExchange


text:
Call for Candidates to AHPA's Board of Trustees
December 2021: Deadlines submission of nominees is February 4, 2022
Published: Wednesday, December 8, 2021
The American Herbal Products Association (AHPA) is issuing its annual call for candidates for elected trustees to the association's Board of Trustees. With the uncertainty of the number and types of in-person events to be held at Expo West in Anaheim, the Board of Trustees has made the decision that the election for the 2022 cycle will again be held in a virtual format during AHPA's annual member meeting, which is currently planned to be held in a hybrid in-person/virtual manner on March 9, 2022. Candidates should be prepared to participate in an online platform where they will have the opportunity to give a short address to the membership.
The Association's bylaws and election policies will govern this nomination process and election.
"AHPA's trustees provide invaluable individual experience and expertise to the staff and to the membership at large, and we value their leadership," said AHPA President Michael McGuffin. "We encourage all eligible companies to nominate a representative and take advantage of this opportunity to become involved in helping to shape the future of this industry."
A candidate for the board may be an owner, employee, consultant, or other designated representative of the company wishing to be represented on the board. A candidate must be officially nominated in writing by completing and returning an Intent to Stand for Election form.
The number of seats on the board of trustees consists of a minimum of 22. Two elected seats are reserved for "small business members," defined as those Active Members with $5 million or less in annual herbal revenues. The remaining elected seats are available to all Active Members in good standing. The term of office for all elected trustees is three years.
The board meets three times each year (including in-person meetings at Natural Products Expo West and SupplySide West trade shows), and trustees are expected to attend all meetings. In addition, trustees are expected to participate in ongoing functions of the organization. While trustees receive no remuneration for their services, the benefits to the companies represented, and the industry as a whole, are well recognized.
The deadline to submit your intent to stand for election form is Friday, February 4, 2022. All active AHPA members in good standing are qualified and encouraged to nominate a candidate.
If you have any questions about this process, please contact Stan Cowan, 301.588.1171 x102.


text:
Manitoba First Nation seeking help from United Nations over their lack of clean water
Darrell Stranger
Long term drinking water advisories remain an issue in many First Nation communities throughout Canada.
Tataskweyak Cree Nation (TCN) in northern Manitoba says it is calling on the United Nations to help pressure the federal government into taking action to fix their drinking water.
The community of 2,300 located about 800 km due north of Winnipeg, has been under a drinking water advisory since 2017, one of many communities throughout Canada that is under a drinking water advisory.
"This issue that we've been dealing with in our community, we can longer ignore it anymore. We have to fight for our people. We matter like all other Canadians right," said TCN Chief Doreen Spence.
In 2019 the community launched a class action lawsuit and that action was certified in July of 2020. The class can include any member of a band whose land was subject to a water advisory that lasted at least one year, at any point from Nov. 8, 1995, until present.
Now, the community is calling on the United Nations to help with their water crisis. The community has written a letter to the UN special rapporteurs.
"Canada's inaction has left TCN no choice to bring this to the United Nations. You know our children, our members are still getting sick from the water and I feel that, you know it's Canada's responsibility and duty to provide to all first nations and clean drinking is one of them," said Spence.
Rapporteurs conduct fact-finding missions to countries to investigate allegations of human rights violations.
The community often buys bottled water because they don't trust the water in the area. Those who do drink the water often develop skin problems such as rashes.
NDP MP Niki Ashton hopes this call for international help spurs the federal government into action.
"If they won't be moved by their own failure, if they won't be moved by a class action lawsuit, hopefully being shamed on the international stage will force them to act. Let's be clear. This is a pattern of this government," Ashton said. "They've been sued for lack of funding for First Nations children in care, over their failure to get clean drinking water.
"It begs the question, how many lawsuits will it take for the federal government to respect their human rights obligations to First Nations.
Tataskweyak conducted two water studies at their own expense in 2018 and 2019 for the water they get from Split Lake.
According to the community's lawyer Michael Rosenberg, the results showed the water treatment plant is unable to treat the various bacteria in the water.
These bacteria include E. coli contamination and three types of cyanobacteria, better known as blue-green algae which produce toxins harmful to humans.
According to the Centers for Disease Control and Prevention, cyanobacteria can create their own toxins called cyanotoxins. These toxins are among the most powerful natural poisons known.
"The water treatment plant is not designed to remove all of those toxins and as Ms. Ashton mentioned, current testing regime does not test for all of them and so part of the effort has been to work with the federal government to put in place a testing regime that will address the concerns that are raised by the studies that Tataskweyak Cree Nation commissioned," Rosenberg said.
As stated on their website, Indigenous Services Canada (ISC), as of Jan. 26 there are currently 57 long term drinking water advisories in 39 communities.
"Indigenous Services Canada (ISC) has supported the community in the repairs and upgrades to their water treatment centre to ensure water quality continues to meet approved guidelines," said a statement from spokesperson Adrienne Vaupshas.
"Since 2016, ISC has provided over $23.5 million toward water and wastewater upgrades, which has provided a new lagoon, lift station, distribution lines, and repairs and upgrades to the water treatment plant, as well as a detailed source water study. A further investment of approximately $700,000 is planned for 2020-2021 to complete water and wastewater work.
"We respect the right of Indigenous groups, including Tataskweyak Cree Nation, to seek intervention of the Courts. We cannot further comment as the matter is before the courts."
Reporter / Winnipeg
Darrell is a proud member of Peguis First Nation in Manitoba. He is a graduate of the television program from Northern Alberta Institute of Technology in Edmonton. He is returning to APTN after having completed an internship with us in 2018 and a brief stop as a reporter in B.C. in 2019.
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