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

TL;DR

The paper constructs a special differentiable, locally Lipschitz function in Euclidean space whose subdifferential sets can represent any convex body, revealing complex behaviors of such functions and their subdifferentials.

Contribution

It introduces a method to realize any convex body as a subdifferential set of a carefully constructed differentiable, locally Lipschitz function, highlighting differences from continuously differentiable functions.

Findings

Any convex body can be realized as a subdifferential set of a differentiable, locally Lipschitz function.

The class of such functions is infinite-dimensional and dense in the space of all locally Lipschitz functions.

The technique can recover all compact connected sets with nonempty interior as subdifferential sets.

Abstract

We construct a differentiable locally Lipschitz function $f$ in $\mathbb{R}^{N}$ with the property that for every convex body $K\subset \mathbb{R}^N$ there exists $\bar x \in \mathbb{R}^N$ such that $K$ coincides with the set $\partial_L f(\bar x)$ of limits of derivatives $\{Df(x_n)\}_{n\geq 1}$ of sequences $\{x_n\}_{n\geq 1}$ converging to~$\bar x$. 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.