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

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)\}_...

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Bibliographic Details
Published in:arXiv.org 2024-09
Main Authors: Daniilidis, Aris, Deville, Robert, Tapia-Garcia, Sebastian
Format: Article
Language:English
Subjects:
Convergence
Vector spaces
Online Access:Get full text
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Summary: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.
ISSN:2331-8422