Representations of epi-Lipschitzian sets

TL;DR

This paper characterizes epi-Lipschitzian sets in Banach spaces as level sets of locally Lipschitz functions with non-zero Clarke gradients at boundary points, extending finite-dimensional results and resolving an open question.

Contribution

It provides a new characterization of epi-Lipschitzian sets in Banach spaces using Clarke's generalized gradient, generalizing finite-dimensional results.

Findings

Characterization of epi-Lipschitzian sets in Banach spaces.

Extension of finite-dimensional results to infinite dimensions.

Resolution of a standing open question in the field.

Abstract

A closed subset $M$ of a Banach space $E$ is \ep, i.e., can be represented locally as the epigraph of a Lipschitz function, if and only if it is the level set of some locally Lipschitz function $f: E\to \R$, wich Clarke's generalized gradient does not contain 0 at points in the boundary of $M$, i.e., such that: M=\{x \mid f(x)\leq 0\}, 0\not \in \partial f(x) {if} x\in \bd M. This extends the characterization previously known in finite dimension and answers to a standing open question