The Engulfing Property from a Convex Analysis Viewpoint

TL;DR

This paper offers a straightforward proof that differentiability and strict convexity are inherent to convex functions with the engulfing property, using only fundamental convex analysis results.

Contribution

It provides a simple, convex analysis-based proof linking the engulfing property to differentiability and strict convexity, clarifying their intrinsic relationship.

Findings

Differentiability is intrinsic to the engulfing property.

Strict convexity is intrinsic to the engulfing property.

The proof relies solely on basic convex analysis results.

Abstract

In this note we provide a simple proof of some properties enjoyed by convex functions having the engulfing property. In particular, making use only of results peculiar to convex analysis, we prove that differentiability and strict convexity are conditions intrinsic to the engulfing property.