Generalized $\mathcal{C}$ -concave conditions and their applications

Abstract

We first introduce two general \(\mathcal{C}\)-concave conditions, and show the implications between \(\mathcal{C}\)-concave, diagonally \(\mathcal{C}\)-concave, diagonally \(\mathcal{C}\)-quasiconcave, and γ-diagonally \(\mathcal{C}\)-quasiconcave conditions which generalize both concavity and quasiconcavity simultaneously without assuming the linear structure. Using the γ-diagonal \(\mathcal{C}\)-quasiconcavity, we prove two non-compact minimax inequalities in a topological space which generalize Fan’s minimax inequality and its generalizations in several aspects. As applications, we will prove a general minimax theorem and basic geometric formulations of the minimax inequality in a topological space.