Let $\lVert\cdot \rVert$ be any norm on $\mathbb{R}^n$ and let $x\in \mathbb{R}^n$ be a point. Let $\textrm{ext}(B_{\lVert x\rVert})$ denote the set of all extremal points of $B_{\lVert x \rVert}:=\{y\in \mathbb{R}^n : \lVert x-y \rVert \leq \lVert x\lVert \} $. Suppose that there is a point $z\in \mathbb{R}^n$ such that
$$\lVert x-se\rVert=\lVert z-se\rVert $$
for all real numbers $s\in [0,1]$ and all $e\in\textrm{ext}(B_{\lVert x\rVert})$.
Is it true that $x=z$?
In all possible examples I was able to imagine, this turned out to be the case. Unfortunately, I cannot think of a structured way to prove that the equality holds in general.