Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$.
Motivation: A lot! For example, in game theory $S$ can be a set of possible (fair) allocations, and we need to suggest a natural method to choose one.
Discussion. If S consists of 2 points, or line in $\mathbb R^2$, we have no natural way to select a center, thus $S$ should be convex and weakly compact. To define these properties, we need vector space and topology, thus the natural setting is Bahach space. If $H$ is $\mathbb R^n$, the natural choice is centroid (center mass), but to define it for general case, we need a natural notion of "uniform density" in a Banach space. Is this someting standard which I do not know? My main example is $H = L^1$, space of all intergable functions $[0,1]\to \mathbb R$. In this case if $S$ consists of all functions with range in $[a,b]$, the center should naturally be a constant function $f(x)\equiv (a+b)/2$. Also, $O$ should be tractable to compute, at least for such a simple examples of $H$ and $S$. A good axiomatic foundation ($O$ is the unique point saisfying axioms A1, A2, and A3) would be a plus.