I encounter a problem related to convex and compact set, which is stated as follows.
Whether or not the following claim is correct?
Claim: Let $C$ be an arbitrary subset of $R^n$ such that $C$ is convex and compact. For each $i = 1, \cdots, n,$ for any point in $C$, denote the largest coordinate for the $i$-th dimension by $e_{\max}^i$ and the smallest coordinate by $e_{\min}^i$. Then, the point $x = [\frac{1}{2}(e_{\max}^1+e_{\min}^1), \cdots, \frac{1}{2}(e_{\max}^n+e_{\min}^n)]^T$ belongs to $C$.
Thanks a lot!
It is in general not valid.
Let $$C = \{ x\in\mathbb R^3 \mid x_1+x_2+x_3 = 1, x \ge 0 \}$$ be the standard simplex in $\mathbb R^3$. Then, $e^i_\min = 0$ and $e^i_\max = 1$ for $1\le i \le 3$. However $$ x^* = \frac{1}{2}(1, 1, 1)^T \not\in C. $$