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The problem: Let $\mathcal{K}\subset R^d$ be a closed and bounded set. $\mathcal{K}$ is a convex set if and only if $\forall x\in R^d, |P_{\mathcal{K}}(x)| =1$ (always exist a unique projection point).

I know how to solve the sufficient condition, that is when $\mathcal{K} $ is a convex set the projection point existing and unique. But I do not know how to prove the necessary condition.

I hope someone kind can help address my concerns

Zhou_Key_Error
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  • This is something of a classical problem, first proven in 1934, and has been proven dozens of ways since, none of which are particularly containable in an answer here. If we replace $R^d$ by an infinite-dimensional Hilbert space, the question is open, and known as the Chebyshev conjecture. Here's a nice survey paper on the problem, including some proofs that work in finite dimensions (see section 3). Is this an exercise from a book? If so, then it would help to know what results the chapter covers, to find a suitable proof. – Theo Bendit Mar 07 '24 at 15:25
  • @TheoBendit Yes, this question comes from "Introduction to Online Convex Optimization". Thanks for your comment! I'm really appreciate your help! This link show a proof to this problem. – Zhou_Key_Error Mar 07 '24 at 17:25

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