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I propose the axiom that any isometry between two such sets must take the center of one onto the center of the other. This axiom by itself is consistent with the existence of a center for every weakly compact convex set by the Ryll-Nardzewski fixed point theorem (we need the group of isometries of $S$ onto itself to always have a fixed point), and it alone already uniquely determines the center in some cases. For example, let $S$ be the positive part of the unit ball of $l^p({\bf Z})$ for $1 \leq < p < \infty$. Translation (taking the sequence $(a_n)$ to the sequence $(a_{n-1})$) is an isometry of this set onto itself, and the only fixed point is the origin.

This example shows that the center will sometimes be an extreme point, which may be counterintuitive. But it's actually reasonable if you look at the center of mass of the positive part of the unit ball of $l^p_n$; as $n \to \infty$ it does converge to the origin (weakly, regarding $l^p_n$ as sitting inside $l^p$).
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I propose the axiom that any isometry between two such sets must take the center of one onto the center of the other. This axiom by itself is consistent with the existence of a center for every weakly compact convex set by the Ryll-Nardzewski fixed point theorem (we need the group of isometries of $S$ onto itself to always have a fixed point), and it alone already uniquely determines the center in some cases. For example, let $S$ be the positive part of the unit ball of $l^p({\bf Z})$ for $1 \leq p < \infty$. Translation (taking the sequence $(a_n)$ to the sequence $(a_{n-1})$) is an isometry of this set onto itself, and the only fixed point is the origin.

This example shows that the center will sometimes be an extreme point, which may be counterintuitive. But it's actually reasonable if you look at the center of mass of the positive part of the unit ball of $l^p_n$; as $n \to \infty$ it does converge to the origin .(weakly, regarding $l^p_n$ as sitting inside $l^p$).
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I propose the axiom that any isometry between two such sets must take the center of one onto the center of the other. This axiom by itself is consistent with the existence of a center for every weakly compact convex set by the Ryll-Nardzewski fixed point theorem (we need the group of isometries of $S$ onto itself to always have a fixed point), and it alone already uniquely determines the center in some cases. For example, let $S$ be the positive part of the unit ball of $l^p({\bf Z})$ for $1 \leq p < \infty$. Translation (taking the sequence $(a_n)$ to the sequence $(a_{n-1})$) is an isometry of this set onto itself, and the only fixed point is the origin.

This example shows that the center will sometimes be an extreme point, which may be counterintuitive. But it's actually reasonable if you look at the center of mass of the positive part of the unit ball of $l^p_n$; as $n \to \infty$ it does converge to the origin.