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

2024/05/15 by Daniilidis, Aris, Deville, Robert, Tapia-Garcia, Sebastian
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2405.09206

Abstract

We construct a differentiable locally Lipschitz function f in ℝ^N with the property that for every convex body K⊂ ℝ^N there exists x ∈ ℝ^N such that K coincides with the set ∂_L f( x) of limits of derivatives \Df(x_n)\_{n≥ 1} of sequences \x_n\_{n≥ 1} converging to~ 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.

Related

• Lipschitz Continuity of Convex Functions
• Characterization of Lipschitz continuous DC functions
• Algebraic Lipschitz and Subdifferential Calculus in General Vector\n Spaces
• Classification of Lipschitz derivatives in terms of semicontinuity and the Baire limit functions
• Locally Lipschitz Functions and Bornological Derivatives