User bogdan - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T11:52:15Z http://mathoverflow.net/feeds/user/24448 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99597/a-natural-center-of-a-convex-weakly-compact-set-in-banach-space A natural center of a convex weakly compact set in Banach space. Bogdan 2012-06-14T11:56:13Z 2012-06-15T14:43:58Z <p>Question: Let S be convex weakly compact set in Banach space H. Propose a natural way to define the unique center O \in S.</p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/99597/a-natural-center-of-a-convex-weakly-compact-set-in-banach-space/99640#99640 Comment by Bogdan Bogdan 2012-06-15T08:26:11Z 2012-06-15T08:26:11Z Yes, this axiom makes sence. However, the non-uniqueness is very serious. Actually, for generic convex compact set S in R^2 there is no nontrivial isometry preserving S, thus any point can be a center! Theorem 1 in Teck-Cheong Lim: The center of a convex set (1981) <a href="http://www.ams.org/journals/proc/1981-081-02/S0002-9939-1981-0593489-7/S0002-9939-1981-0593489-7.pdf" rel="nofollow">ams.org/journals/proc/1981-081-02/&hellip;</a> defines the unique center, and it satisfies your axiom. What I do not like is that 1) this construction uses transivite induction and thus can be hardly applied on practice; 2) in case of R^n it does not coincide with centroid. http://mathoverflow.net/questions/99597/a-natural-center-of-a-convex-weakly-compact-set-in-banach-space Comment by Bogdan Bogdan 2012-06-15T08:10:08Z 2012-06-15T08:10:08Z For my applications, it is absolutely crutial for the center to belong to the set. This is not the case in Theorem A you mention, as well as for Chebyshev center...