# The minimal face of a polytope containing a set

Encountered the following statement while reading a paper where it was stated without proof - am wondering why its true.

Suppose $P$ is a polytope, $M$ is a convex subset of $P$. Define $f(M)$ to be the minimal face of $P$ which contains $M$. Then there is a point in $M$ which is in the relative interior of $f(M)$.

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You should consider first the case where $M =\{x\}$ is a single point. Then $x$ must lie in the relative interior of $f(M)$ since otherwise it would lie on a proper face of $f(M)$, contradicting the definition of a minimal face (a definition which you might want to state explicitly). Now in considering more general $M$, you should realize that the convexity assumption is essential (consider $P$ a tetrahedron, and $M$ consisting of two incident edges). – yasmar Jul 14 '12 at 12:28
Thanks - yes, I see, $M$ needs to be convex indeed. – atricks Jul 14 '12 at 14:53
Could you add details? I don't see why $x$ has to lie in the relative interior of $f(M)$ even if $M=\{x\}$ is a single point. – atricks Jul 14 '12 at 14:53
@atricks: Because the exterior of a face is formed by smaller faces, if $x$ were to lie in the exterior of $f(M)$, it would also lie in a smaller face, so $f(M)$ wouldn't be minimal. – joriki Jul 14 '12 at 20:31
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