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Given $L,K$, two compact and convex set, with $L, K \subseteq R^p$.

Their support functions will be, $supp(K,p): R^p \to R$ and $supp(L,p):R^p \to R$, respectively.

In the article "$L_p$ Metrics for Compact, Convex Sets" of Richard Vitale of $1984$ I found the following property: $$ K \subseteq L \iff supp(K,p) \leq supp(L,p) $$

I tried to look for a proof of this statement but I have not been able to find anything.

Emilio Costa
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  • Did you try proving it yourself? Where did you get stuck? – littleO Oct 07 '18 at 10:57
  • At the moment, I'm just approaching the convex-analysis and I'm still trying to understand how it works, but the main problem is proving that if the $supp(K) \leq supp(L)$ then $K \subseteq L$. – Emilio Costa Oct 07 '18 at 11:12
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    Why is this question getting close votes? It seems like a good question. – littleO Oct 07 '18 at 11:26
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    I thought this was an interesting question. OP mentioned that he found the statement in a research paper and that the main difficulty was in proving the converse, which seems like enough context to me. I learned something from solving it and I'd like to see if anyone else has a simpler solution. – littleO Oct 07 '18 at 20:52

1 Answers1

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Background knowledge: The convex conjugate of a convex function $f$ is denoted by $f^*$: $$ f^*(z) = \sup_x \, \langle z, x \rangle - f(x). $$ We can see that $$ f \leq g \implies f^* \geq g^*. $$ Also recall that if $f$ is a proper closed convex function then $$ f^{**} = f. $$

Now we can answer the question. Let $I_K$ and $I_L$ be the convex indicator functions of $K$ and $L$, respectively, and let $S_K$ and $S_L$ be the support functions of $K$ and $L$ respectively. Notice that $I_K^* = S_K$ and $S_K^* = I_K^{**} = I_K$. With this notation in place we are ready to give our proof: $$ K \subset L \implies I_K \geq I_L \implies I_K^* \leq I_L^* \implies S_K \leq S_L. $$ Conversely, $$ S_K \leq S_L \implies S_K^* \geq S_L^* \implies I_K \geq I_L \implies K \subset L. $$

littleO
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