Prove that for $p \ge 1$, function $f(x) = |d_C (x)|^p, x \in \mathbb{R}^n$ is a convex function

by Minh Nguyễn Hoàng   Last Updated July 12, 2019 07:20 AM

Question: Let $C \subset \mathbb{R}^n$ be a convex set, $d_C (x)$ is distance from $x \in \mathbb{R}^n$ to $C$. Prove that lower differential of the function $d_C$ at $x$ defined by $$\partial d_c (x) = \{x^* \in \mathbb{R}^n : (x^*, y-x) <0, \forall y \in C \text{ and } ||x^*|| \le 1\}.$$

Could you give me some hint to solve this problem. Thank all!

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