Let $(X,\|\cdot\|)$ be a finite dimensional normed space and $A\subset X$ be a convex set with nonempty interior and let $extr(A)=\{e_1,e_2,\ldots e_m\}$ (extr(A)={the set of all extreme points of $A$}). Suppose further there exist $x_0\in X$ such that $\|x_0 - e_i\|=c$ for every $i\in \{1,2,\ldots m\}$, where $c-$some positive fixed number. Must the point $x_0$ lie in $A$, i.e. is it true that $x_0\in A$?
Any help is very appreciated.
In general, $x_0$ is not in $A$. To find a counterexample, fix $x_0$ and consider the sphere $S$ of point a fixed distance $r>0$ away from $x_0$. Then, you can place some points $e_i$ on $S$ such that $x_0$ doesn't lie in the convex hull of $\{e_1,\ldots,e_n\}$.
For example, on $\mathbb R^2$ with the euclidean metric, consider $x_0 = (0,0)$ and : \begin{align} e_1 &= (1,0) \\ e_2 &= (1/\sqrt{2},1/\sqrt{2}) \\ e_3 &= (1/\sqrt{2},-1/\sqrt{2}) \end{align}