Differentiability of cone-monotone functions on separable Banach space

Abstract

Abstract. Motivated by applications to (directionally) Lipschitz functions, we provide a general result on the almost everywhere Gâteaux differentiability of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone with nonempty interior. This seemingly arduous restriction is useful, since it covers the case of directionally Lipschitz functions, and necessary. We show by way of example that most results fail more generally.

Cone-Monotone Functions: Differentiability and Continuity

“…If f : X → R ∪ {+∞} is K-monotone, then f is Gâteaux differentiable a.e. [2]. As shown in Borwein and Goebel [3], if K has empty interior, almost anything can happen for K-monotone functions.…”

“…More pathological examples concerning K-monotone functions when K has empty interior can be found in [2,3].…”

“…Cone-monotone functions have been considered by Ward, Chabrillac-Crouzeix, and Saks on R n [4,10], Borwein, Burke, and Lewis [2] on separable spaces -for K having non-empty interior. The key positive result is: Suppose X is separable and K ⊂ X is a convex cone with non-empty interior.…”

“…In Section 2, we illustrate that the class of conemonotone functions is significantly broad; it includes Lipschitz functions, quasiconvex functions, and marginal value functions. In Section 3, we give an alternative proof to the differentiability theorem of cone-monotone functions on separable Banach spaces (due to Borwein, Burke and Lewis [2]) using the notion of σ-directionally porous sets. Section 4 deals with continuity, measurability, and extendibility of conemonotone functions.…”

Haar null and Haar meager sets: a survey and new results

“…The Rademacher theorem states that a Lipschitz function between Euclidean spaces is differentiable almost everywhere. The second motivation of Christensen was to extend this result to Banach spaces, see the paper [24] by Christensen, and also [15–18, 90].…”