Representations of epi-Lipschitzian sets

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 $f:E\to \mathbb{R}$, 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