Differentiability of quasiconvex functions on separable Banach spaces

TL;DR

This paper studies the differentiability of quasiconvex functions on separable Banach spaces, showing that under mild conditions, such functions are differentiable on large dense subsets, extending classical results to infinite-dimensional settings.

Contribution

It extends known differentiability results for quasiconvex functions from finite-dimensional spaces to separable Banach spaces, under weaker assumptions.

Findings

Quasiconvex functions are Hadamard differentiable on dense subsets if usc or strictly quasiconvex.

Even quasiconvex functions are Gateaux differentiable on dense subsets and continuous.

Results include that in reflexive spaces, the differentiability set complements a Haar null set.

Abstract

We investigate the differentiability properties of real-valued quasiconvex functions f defined on a separable Banach space X. Continuity is only assumed to hold at the points of a dense subset. If so, this subset is automatically residual. Sample results that can be quoted without involving any new concept or nomenclature are as follows: (i) If f is usc or strictly quasiconvex, then f is Hadamard differentiable at the points of a dense subset of X (ii) If f is even, then f is continuous and Gateaux differentiable at the points of a dense subset of X. In (i) or (ii), the dense subset need not be residual but, if X is also reflexive, it contains the complement of a Haar null set. Furthermore, (ii) remains true without the evenness requirement if the definition of Gateaux differentiability is generalized in an unusual, but ultimately natural, way. The full results are much more general and substantially stronger. In particular, they incorporate the well known theorem of Crouzeix, to the effect that every real-valued quasiconvex function on R^N is Frechet differentiable a.e.