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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Bibliographic Details
Published in:Proceedings of the London Mathematical Society Vol. 129; no. 5
Main Authors: Daniilidis, Aris, Deville, Robert, Tapia‐García, Sebastián
Format: Journal Article
Language:English
Published: London Mathematical Society 01.11.2024
Subjects:
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
Online Access:Get full text
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Summary: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