Rotund norms, Clarke subdifferentials and extensions of Lipschitz functions
It is well known that the Clarke subdifferential of a Lipschitz function is an essential tool in nonsmooth analysis and especially in nonsmooth optimization. Nevertheless, there exists a large class of pathological Lipschitz functions for which this notion provides no information about the local behavior of the function. So in recent papers it is shown that the set of 1-Lipschitz functions on any Banach space \(X\) for which \(\partial_C f(x)= B_{X^*}\) for all \(x\in X\), is residual (i.e., the complement of a set of first category) in the set of all 1-Lipschitz functions on \(X\).NEWLINENEWLINENEWLINEIn the present paper, the authors discuss the following question:NEWLINENEWLINENEWLINEGiven any closed proper subspace \(Y\subset X\) and any bounded 1-Lipschitz function \(f: Y\to\mathbb{R}\), is there a bounded 1-Lipschitz function \(f: X\to\mathbb{R}\) such that \(\widetilde f|_Y= f\) and \(\partial_C f(x)= B_{X^*}\) for all \(x\in X\)?NEWLINENEWLINENEWLINEThough the answer can be negative for finite-dimensional spaces with special norm (here a simple example is presented) it is shown that there are many Banach spaces for which the answer is positive. Such spaces are specified by the so-called \textit{maximal subdifferential extension property} (MSEP).NEWLINENEWLINENEWLINEIn the main theorem of the paper it is shown that the MSEP is closely related to special separation properties of \(X\) regarding Lipschitz functions: the so-called Lipschitz separation property and the densely Lipschitz separation property. These properties can be characterized further in terms of different rotundity properties to the unit sphere or to the space \(X\), respectively. (The Banach space \(X\) is called to be rotund if for points \(x\), \(y\) of the unit sphere the relation \(\|x+y\|= 2\) implies \(x= y\).) For finite-dimensional Banach spaces all introduced notions are equivalent. For infinite spaces however (even for reflexive Banach spaces) these equivalences fail in general.
scientific article; zbMATH DE number 1693157