Convex defining functions for convex domains

TL;DR

This paper presents three proofs demonstrating that smoothly bounded convex domains in R^n possess smooth defining functions with Hessians that are non-negative definite near the boundary.

Contribution

The paper provides multiple proofs establishing the existence of smooth defining functions with convex Hessians for convex domains.

Findings

Existence of smooth defining functions with convex Hessians near the boundary

Three different proofs of the main fact

Reinforcement of the relationship between convexity and defining functions

Abstract

We give three proofs of the fact that a smoothly bounded, convex domain in R^n has smooth defining functions whose Hessians are non-negative definite in a neighborhood of the boundary of the domain.