Let $\mathcal{U}$ be a convex and compact set. Its support function is defined as $\delta^*(v|\mathcal{U})=\sup_{u\in \mathcal{U}} v^T u$. Assume that we are given the support function $\delta^*(v|\mathcal{U})$ of a set $\mathcal{U}$, and assume that $\delta^*(v|\mathcal{U})$ is a closed, finite-valued, positively homogenous convex functions (which ensures a bijection between $\delta^*$ and $\mathcal{U}$). How can I find the corresponding set $\mathcal{U}$?
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$\begingroup$ $\mathcal{U}$ would be the intersection of the hyperplanes $H_v=\{u\in\mathbb{R}^n;v^tu\leq\delta^*(v|\mathcal{U})\}$ for each $v$. If the compact $\mathcal{U}$ is arbitrary I doubt there is going to be a simpler way to express it. The proof that this intersection is exactly the set you look for can be completed using, for example, theorem 3.4 of Rudin´s $\textit{Functional Analysis}$, which implies that given a compact convex set and a point outside it you can separate them by a hyperplane. $\endgroup$– Saúl RMCommented Dec 9, 2021 at 12:42
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$\begingroup$ Sorry, I meant half spaces, not hyperplanes. $\endgroup$– Saúl RMCommented Dec 9, 2021 at 12:49
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$\begingroup$ I agree with @SaúlRodríguezMartín. There is not simple way in general. You need to do the computations by hand… $\endgroup$– DirkCommented Dec 9, 2021 at 13:09
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