Linear structure of functions with maximal Clarke subdifferential

TL;DR

This paper proves that the set of Lipschitz functions with maximal Clarke subdifferential forms a large linear space, constructed explicitly without relying on Baire category, extending previous results on the structure of such functions.

Contribution

It provides a constructive proof that the set of Lipschitz functions with maximal Clarke subdifferential contains an uncountably dimensional linear subspace, advancing the understanding of their linear structure.

Findings

The set of functions with maximal Clarke subdifferential contains an uncountably dimensional linear subspace.

The approach is constructive and does not depend on Baire category theorem.

The set is spaceable within the space of all Lipschitz functions.

Abstract

It is hereby established that the set of Lipschitz functions $f:\mathcal{U}\rightarrow \mathbb{R}$ ($\mathcal{U}$ nonempty open subset of $\ell_{d}^{1}$) with maximal Clarke subdifferential contains a linear subspace of uncountable dimension (in particular, an isometric copy of $\ell^{\infty}(\mathbb{N})$). This result goes in the line of a previous result of Borwein-Wang. However, while the latter was based on Baire category theorem, our current approach is constructive and is not linked to the uniform convergence. In particular we establish lineability (and spaceability for the Lipschitz norm) of the above set inside the set of all Lipschitz continuous functions.