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

We construct a differentiable locally Lipschitz function f$f$ in RN$\mathbb {R}^{N}$ with the property that for every convex body K⊂RN$K\subset \mathbb {R}^N$ there exists x¯∈RN$\bar{x} \in \mathbb {R}^N$ such that K$K$ coincides with the set ∂Lf(x¯)$\partial _L f(\bar{x})$ of limits of derivatives...

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Vydáno v:	Proceedings of the London Mathematical Society Ročník 129; číslo 5
Hlavní autoři:	Daniilidis, Aris, Deville, Robert, Tapia‐García, Sebastián
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	London Mathematical Society 01.11.2024
Témata:	Classical Analysis and ODEs Functional Analysis Mathematics Optimization and Control Differentiable Lipschitz function subdifferential range Differentiable Lipschitz function subdifferential range convex body spaceability AMS Subject Classification Primary 26A30 49J52 49J52 Secondary 26A24 26A16 26A27 spaceability spaceability AMS Subject Classification Primary 26A30 Secondary 26A24 26A16 26A27 convex body
ISSN:	0024-6115, 1460-244X
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Popis
Shrnutí:	We construct a differentiable locally Lipschitz function f$f$ in RN$\mathbb {R}^{N}$ with the property that for every convex body K⊂RN$K\subset \mathbb {R}^N$ there exists x¯∈RN$\bar{x} \in \mathbb {R}^N$ such that K$K$ coincides with the set ∂Lf(x¯)$\partial _L f(\bar{x})$ of limits of derivatives {Df(xn)}n⩾1$\lbrace Df(x_n)\rbrace _{n\geqslant 1}$ of sequences {xn}n⩾1$\lbrace x_n\rbrace _{n\geqslant 1}$ converging to x¯$\bar{x}$. 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:	0024-6115 1460-244X
DOI:	10.1112/plms.70007