Representations of epi-Lipschitzian sets

A closed subset M of a Banach space E is epi-Lipschitzian, 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→R, for which Clarke's generalized gradient does not contain 0 at points in the boundary of M, i.e., such that: M={x∣f(x)≤0},0∉∂f(x) if x∈bdM. This extends the characterization previously known in finite dimension and answers to a standing question.