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



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> **Question:** Is there a point $x\in X$ fixed by all $\phi\in\operatorname{Isom}(X)$?
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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?](https://math.stackexchange.com/questions/4449267/a-version-of-browers-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 $\Bbb R^n$](https://mathoverflow.net/questions/422384/symmetries-of-contractable-subsets-of-bbb-rn).



Does a compact contractible metric space have a point that is fixed by all isometries?
[There are](https://mathoverflow.net/questions/78501/do-finite-groups-acting-on-a-ball-have-a-fixed-point) 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](https://en.wikipedia.org/wiki/Mostow%E2%80%93Palais_theorem%20), 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.


