On some convexities

Abstract

The aim of the present paper is to investigate some convexities. First we introduce and investigate the "pseudoconvex" spaces and give a minimax theorem applying the ideas of [8]. The notion of convex spaces was introduced by Komiya [1] (see below for details). In [2] the authors proved a Nikaidd--Isoda-type theorem for convex spaces. An interesting convexity notion, not giving a convex space in the sense of [1] is obtained in [5]. The paper [4] contains investigations with respect to this convexity structure, analogous to that of in [2]. Now we give a common generalization of the convexities of [i] and [5], the so-called pseudoconvex space. It turns out that the compact pseudoconvex spaces have the fixed point property and a Nikaido-Isoda-type theorem holds. First we prove these assertions and after we prove an inequality between the Helly and Caratheodory number of pseudoconvex spaces. In connection with these investigations we mention the work of M. Horv~th [3] where the Helly, Caratheodory and Radon numbers of a special convexity structure are calculated and whose proof is based on graph theoretical results. At last we continue the investigation of V. Komornik [10] and give a related minimax theorem in interval space.