TITLE: Does a compact contractible metric space have a point that is fixed by all isometries?
QUESTION [10 upvotes]: 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 $\Bbb R^n$.

REPLY [15 votes]: There are finite groups that act smoothly on a disk without a global fixed point. You can arrange the metric to be isometric, e.g. via the Mostow-Palais embedding theorem, which equivariantly and smoothly embeds the disk into a Euclidean space where the group acts orthogonally, and taking the metric induced by Euclidean one. The full isometry group is potentially bigger, so it cannot fix a point.