All convex bodies are in the subdifferential of some everywhere differentiable locally Lipschitz function - Daniilidis, Aris; Deville, Robert; Tapia‐García, Sebastián (Proceedings of the London Mathematical Society, 2024)
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 xnn⩾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.