Geometric characterizations of the strict Hadamard differentiability of sets

TL;DR

This paper characterizes strict Hadamard differentiability of sets and functions in Banach spaces using geometric properties of tangent cones, providing necessary and sufficient conditions involving hyperplanes and isomorphisms.

Contribution

It offers new geometric characterizations of strict Hadamard differentiability for sets and Lipschitz functions in Banach spaces, linking differentiability to tangent cone structures.

Findings

Differentiability at boundary points relates to hyperplanes in tangent cones.

Strict Hadamard differentiability is characterized by isomorphisms of tangent cones.

Continuity of set-valued mappings is key to differentiability conditions.

Abstract

Let $S$ be a closed subset of a Banach space $X$. Assuming that $S$ is epi-Lipschitzian at $\bar{x}$ in the boundary $ \bd S$ of $S$, we show that $S$ is strictly Hadamard differentiable at $\bar{x}$ IFF the Clarke tangent cone $T(S, \bar{x})$ to $S$ at $\bar{x}$ contains a closed hyperplane IFF the Clarke tangent cone $T(\bd S, \bar{x})$ to $\bd S$ at $\bar{x}$ is a closed hyperplane. Moreover when $X$ is of finite dimension, $Y$ is a Banach space and $g: X \mapsto Y$ is a locally Lipschitz mapping around $\bar{x}$, we show that $g$ is strictly Hadamard differentiable at $\bar{x}$ IFF $T(\mathrm{graph}\,g, (\bar{x}, g(\bar{x})))$ is isomorphic to $X$ IFF the set-valued mapping $x\rightrightarrows K(\gh g, (x, g(x)))$ is continuous at $\bar{x}$ and $K(\gh g, (\bar{x}, g(\bar{x})))$ is isomorphic to $X$, where $K(A, a)$ denotes the contingent cone to a set $A$ at $a \in A$.