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.
Keywords
Differentiable Lipschitz function
subdifferential range
convex body
spaceability
Differentiable Lipschitz function subdifferential range convex body spaceability AMS Subject Classification Primary 26A30 49J52
Secondary 26A24 26A16 26A27
spaceability AMS Subject Classification Primary 26A30
49J52
Secondary 26A24
26A16
26A27
Origin | Files produced by the author(s) |
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