Vector-Valued Convex Functions

Abstract

Convex analysis plays a fundamental role in mathematics. In this paper, we extend the concept of convexity to vector-valued functions in Banach lattices. We introduce the notion of “order convexity” (o-convexity) and explore its properties, generalizing several results from real-valued convex analysis. These include a continuity theorem for o-convex functions (Theorem 1.2), an analogue of Bauer’s maximal principle for o-convex functions on compact sets (Theorem 2.2), and a fixed-point theorem for order contraction maps (Theorem 2.3).