A primal approach to the Clarke-Ledyaev inequality

TL;DR

This paper introduces a primal, geometry-based approach to the Clarke-Ledyaev inequality, simplifying its proof by avoiding dual space elements and relying on classical geometric lemmas.

Contribution

It provides a new primal proof of the Clarke-Ledyaev inequality, eliminating the need for dual space arguments and enhancing understanding through geometric methods.

Findings

Simplified proof of the Clarke-Ledyaev inequality

Elimination of dual space elements in the proof

Use of classical geometric lemmas

Abstract

We present a version of the Clarke-Ledyaev inequality that does not involve elements of the dual space. The proof relies mainly on geometry and on the classical lemma of Bishop and Phelps. In addition, this approach allows us to provide a simplified proof of the Clakre-Ledyaev inequality. The approach is primal in the sense that no dual arguments are used.