Let $(X,d)$ be a compact and contractible metric space. Let $\mathrm{Isom}(X)=\{\phi\colon X\to X\}$ be its group of isometries.

Question: Is there a point $x\in X$ fixed by all $\phi\in\mathrm{Isom}(X)$?

I am happy to assume some additional niceness conditions for $X$, enough to ensure that $X$ satisfies some fixed-point theorem, guaranteeing that every continuous map $\phi\colon X\to X$ has a fixed point (e.g. triangulable, locally contractible; see here for details). The emphasize is therefore on whether all isometries have a common fixed point.

This post is a refinement/generalization of this question.

Let $(X,d)$ be a compact and contractible metric space. Let $\operatorname{Isom}(X)=\{\phi\colon X\to X\}$ be its group of isometries.

Question: Is there a point $x\in X$ fixed by all $\phi\in\operatorname{Isom}(X)$?

I am happy to assume some additional niceness conditions for $X$, enough to ensure that $X$ satisfies some fixed-point theorem, guaranteeing that every continuous map $\phi\colon X\to X$ has a fixed point (e.g. triangulable, locally contractible; see A version of Brower's fixed point theorem for contractible sets? for details). The emphasize is therefore on whether all isometries have a common fixed point.

This post is a refinement/generalization of Symmetries of contractable subsets of $\Bbb R^n$.