Minimax theorems with applications to convex metric spaces

Abstract

A minimax theorem is proved which contains a recent result of Pinelis and a version of the classical minimax theorem of Ky Fan as special cases. Some applications to the theory of convex metric spaces (farthest points, rendez-vous value) are presented.

Central points and measures and dense subsets of compact metric spaces

“…To show this, we need the next result. It was proved (in a different way) in special cases by Takahashi [18] and Kindler [9]. d) is a weakly convex compact metric space having more than one point, then r(X) < δ(X).…”

“…for all w ∈ X; Kijima [8] and Yang and Zhang [22] speak about convexity when (3-1) with λ = 1 2 is fulfilled; while Kindler [9] says about ϕ-convexity for any concave, nondecreasing in both variables function ϕ such that ϕ(x, y) < max(x, y) whenever x = y. The reader interested in this topic is referred to the original papers of the above mentioned authors.…”

“…Now we shall recall the classical atributes of a metric space (see e.g. [3] or [9]). By δ(X) we denote the diameter of a metric space (X, d), that is, δ(X)…”