On the Properties of Convex Functions over Open Sets

TL;DR

This paper explores the unique properties of smooth convex functions defined on open convex sets, revealing differences from functions defined on entire spaces and introducing a stronger inequality than the Descent Lemma.

Contribution

It demonstrates that smooth convex functions on open sets cannot always be extended to the whole space and derives a new, stronger inequality related to these functions.

Findings

Existence of functions not extendable to entire space while preserving convexity

New inequality stronger than the classical Descent Lemma

Distinct properties of convex functions on open sets

Abstract

We consider the class of smooth convex functions defined over an open convex set. We show that this class is essentially different than the class of smooth convex functions defined over the entire linear space by exhibiting a function that belongs to the former class but cannot be extended to the entire linear space while keeping its properties. We proceed by deriving new properties of the class under consideration, including an inequality that is strictly stronger than the classical Descent Lemma.