Extending Rademacher Theorem to Set-Valued Maps

TL;DR

This paper extends Rademacher's theorem to set-valued maps by introducing a suitable notion of differentiability, broadening the theorem's applicability beyond single-valued functions in Euclidean spaces.

Contribution

It introduces a new concept of set-valued differentiability linked to convex processes, generalizing Rademacher's theorem to set-valued maps and recovering the classical case.

Findings

Extended Rademacher theorem to set-valued maps.

Established a new notion of set-valued differentiability.

Unified classical and set-valued differentiability results.

Abstract

Rademacher theorem asserts that Lipschitz continuous functions between Euclidean spaces are differentiable almost everywhere. In this work we extend this result to set-valued maps using an adequate notion of set-valued differentiability relating to convex processes. Our approach uses Rademacher theorem but also recovers it as a special case.