All convex bodies are in the subdifferential of some everywhere differentiable locally Lipschitz function

Abstract

We construct a differentiable locally Lipschitz function (Formula presented.) in (Formula presented.) with the property that for every convex body (Formula presented.) there exists (Formula presented.) such that (Formula presented.) coincides with the set (Formula presented.) of limits of derivatives (Formula presented.) of sequences (Formula presented.) converging to (Formula presented.). The technique can be further refined to recover all compact connected subsets with nonempty interior, disclosing an important difference between differentiable and continuously differentiable functions. It stems out from our approach that the class of these pathological functions contains an infinite-dimensional vector space and is dense in the space of all locally Lipschitz functions for the uniform convergence.