The fixed point property for weakly admissible compact convex sets: searching for a solution to Schauder's conjecture

Abstract

A compact convex set X in a linear metric space is weakly admissible if for every ε > 0 there exist compact convex subsets X 1,…, X n of X with X = conv( X 1 ∪ … ∪ X n ) and continuous maps ƒ i from X i into finite dimensional subsets E i , i = 1, …, n, of X such that ∑ n i = 1 ∥ƒ i(x i) − x i∥ < ε for every x i ϵ X i , and i = 1, …, n. Theorem: Any weakly admissible compact convex set has the fixed point property. Question: Is every weakly admissible compact convex set an AR?