On Subgradients of Convex Functions and Orlicz Pseudo-Norms for Vector-Valued Functions

TL;DR

This paper develops a unified framework for convex analysis with vector-valued targets by (i) proving the existence of measurable subgradients for multivariate convex functions and linking these subgradients to directional derivatives, (ii) characterizing the $\Delta_2$-condition and its two-sided variant through directional derivatives and associated Young functions, and (iii) extending Luxemburg and Orlicz pseudo-norms to vector-valued functions, including duality relationships, perturbation stability, and concavity properties with respect to measure mixtures. The results provide foundational tools for transport-type energy estimates and multivariate Orlicz space analysis, with explicit criteria for when the dual transform preserves growth conditions and how these norms interact under perturbations and convex combinations. Overall, the work deepens the understanding of vector-valued convex analysis and its implications for functional-analytic approaches in optimization and transport problems.

Abstract

We discuss variants of construction of measurable subgradients for multivariate convex functions and the problem of characterization of the $Δ_2$-condition in terms of their directional derivatives. Furthermore we study related basic properties of Luxemburg and Orlicz pseudo-norms for vector-valued functions.