All convex bodies are in the subdifferential of some everywhere differentiable locally Lipschitz function - Institut de Mathématiques de Bordeaux
Preprints, Working Papers, ... Year : 2024

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

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.
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Dates and versions

hal-04571954 , version 1 (14-05-2024)
hal-04571954 , version 2 (11-09-2024)

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Aris Daniilidis, Robert Deville, Sebastian Tapia-Garcia. All convex bodies are in the subdifferential of some everywhere differentiable locally Lipschitz function. 2024. ⟨hal-04571954v2⟩
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