Lipschitz Modulus of Convex Functions via Function Values

TL;DR

This paper demonstrates how to determine the Lipschitz continuity and constants of convex functions using only their function values, providing a practical approach for analyzing such functions.

Contribution

It introduces a method to compute the Lipschitz modulus of convex functions solely from their function values, including global cases and examples beyond norms.

Findings

Lipschitz constants can be derived from function values alone.

Global Lipschitz modulus exists for certain convex functions.

Examples include classes of convex functions beyond norms.

Abstract

In this note, we establish the Lipschitz continuity of finite-dimensional globally convex functions on all given balls and global Lipschitz continuity for eligible functions of that type. The Lipschitz constants in both situations draw information solely from function values, and the global Lipschitz modulus is found when it exists. Some examples of classes of globally Lipschitz continuous convex functions beside the norms are also provided along with their global Lipschitz modulus.