Fixed points for convex continuous mappings in topological vector spaces (Q2718984)

A classical theorem of Brouwer states that any continuous mapping from compact convex set \(C\) in the space \(\mathbb{R}^n\) into itself has at least a fixed-point, say that \(C\) has the fixed-point property (FPP). Extending Brouwer fixed-point theorem, Schauder and Tikhonov proved that compact convex sets in locally convex spaces have the FPP. Schauder also conjectured that Brouwer's fixed point theorem holds for compact convex sets in non-locally convex spaces. Schauder's conjecture is one of the most challenging problems in the fixed-point theory and is of great interest to many researchers on the field.NEWLINENEWLINENEWLINEThe author of the paper under review introduces the notion of convex continuous mappings. The main result is slightly more general than the following: Any continuous convex mapping from compact convex set \(C\) in a linear metric space into itself has a fixed point. (A mapping \(T: X\to X\) is convex continuous at \(x_0\in X\) if for any open neighborhood \(N(Tx_0)\) of \(Tx_0\) there exists an open neighborhood \(V(x_0)\) of \(x_0\) such that \(\text{Conv}(TV(x_0))\subset N(Tx_0)\), where \(\text{Conv}(A)\) denotes the convex hull of \(A\).)NEWLINENEWLINENEWLINEJAs an application of his theorem the author obtains the following result proved previously by the reviewer and \textit{Le Hoang Tri}: The compact convex set with no extreme points constructed by \textit{J. W. Roberts} [Stud. Math. 60, 255-266 (1977; Zbl 0397.46010)] has the FPP.