Question: Let S be 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 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 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 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.