Convex functions with unbounded gradient

TL;DR

This paper investigates which domains can support convex functions with an unbounded gradient at their boundary, i.e., where $|\nabla f(x)|$ may grow without bound as $x$ approaches $\partial \Omega$. The authors prove that any domain enabling such boundary gradient behavior must itself be convex. The approach is a theoretical analysis in convex analysis that derives a geometric implication for the domain from the boundary gradient condition. This result clarifies the relationship between domain geometry and boundary behavior of convex functions, with implications for understanding boundary phenomena in optimization and convex analysis.

Abstract

We show that domains, that allow for convex functions with unbounded gradient at their boundary, are convex.

Figures (1)

• Figure 1: Geometric situation

Theorems & Definitions (5)

• Theorem 1
• Theorem 3
• proof
• Remark 4
• Remark 5