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Let $B$ stand for a compact convex body in a Hilbert space $H$.

Suppose that a finite group $G=\{g_1,\ldots,g_n\}$ acts, via affine maps, on $B$.

Write $B'$ for the image of $B$ in $B^n$ by the map $b\mapsto (g_1b,\ldots,g_nb)$.

$G$ naturally also then acts on $B'$, the new action affinely conjugate to the old action on $B$, but now $G$ acts on $B'$ isometrically.

Consideration of the group of all the affine symmetries of the Hilbert cube shows that one needs some hypothesis on $G$ to get a conjugate action by isometries. I do suspect one gets a similar result for compact groups via Haar measure.

Question: In general, how weak a hypothesis on $G$ (or on $B$ if $G$ is the full affine symmetry group) will suffice?

Also, is there any literature concerning uniformization of actions of convex sets. For example, I'd be interested to know of any applications even of the observation made above.

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