Generalized mathcal{C}-concave conditions and their applications

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 ??-diagonally $mathcal{C}$ -quasiconcave conditions which generalize both concavity and quasiconcavity simultaneously without assuming the linear structure. Using the ??-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.