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

A compact convex set X in a linear metric space is weakly admissible if for every ε > 0 there exist compact convex subsets X1,…,Xn of X with X = conv(X1 ∪ … ∪ Xn) and continuous maps ƒi from Xi into finite dimensional subsets Ei, i = 1, …, n, of X such that ∑ni = 1 ∥ƒi(xi) − xi∥ < ε for every xi ϵ Xi, and i = 1, …, n. m: Any weakly admissible compact convex set has the fixed point property. on: Is every weakly admissible compact convex set an AR?