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

TL;DR

This paper explores measurable subgradients of multivariate convex functions, characterizes the Δ₂-condition via directional derivatives, and examines properties of Luxemburg and Orlicz pseudo-norms for vector-valued functions.

Contribution

It introduces new insights into subgradient measurability, Δ₂-condition characterization, and properties of Orlicz pseudo-norms in the context of vector-valued functions.

Findings

Characterization of the Δ₂-condition through directional derivatives.

Analysis of properties of Luxemburg and Orlicz pseudo-norms.

Construction methods for measurable subgradients of multivariate convex functions.

Abstract

We discuss variants of construction of measurable subgradients for multivariate convex functions and the problem of characterization of the $\Delta_2$-condition in terms of their directional derivatives. Furthermore we study related basic properties of Luxemburg and Orlicz pseudo-norms for vector-valued functions.