## Return to Answer

2 Added 'convex' for clarification.

If $s$ is a non-zero section whose image lies in $\mathcal U_{n,d}$, then it has constant sign on $V^\ast:=\mathbb R^{n+1}\setminus{0}$ and after possibly multiplying by $-1$ we may assume that $s$ is strictly positive on $V^\ast$. The strictly positive $s$ form an open convex cone $C$ (we do not assume that $0$ belongs to a cone) and is hence contractible when non-empty which this one is when $d$ is even. As $C\to\mathcal U_{n,d}$ is a fibration with fibres $\mathbb R_+$ so is $\mathcal U_{n,d}$.

1

If $s$ is a non-zero section whose image lies in $\mathcal U_{n,d}$, then it has constant sign on $V^\ast:=\mathbb R^{n+1}\setminus{0}$ and after possibly multiplying by $-1$ we may assume that $s$ is strictly positive on $V^\ast$. The strictly positive $s$ form an open cone $C$ (we do not assume that $0$ belongs to a cone) and is hence contractible when non-empty which this one is when $d$ is even. As $C\to\mathcal U_{n,d}$ is a fibration with fibres $\mathbb R_+$ so is $\mathcal U_{n,d}$.
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