Sigma-porosity is separably determined

TL;DR

This paper establishes a separable reduction theorem for sigma-porosity of Suslin sets in Banach spaces, enabling the extension of differentiability results for Lipschitz functions from separable to nonseparable spaces.

Contribution

It introduces a method to reduce sigma-porosity questions to separable subspaces using elementary submodels, broadening applicability in nonseparable Banach spaces.

Findings

Proves sigma-porosity is separably determined for Suslin sets.

Extends differentiability theorems to nonseparable Asplund spaces.

Provides a unified approach using elementary submodels.

Abstract

We prove a separable reduction theorem for sigma-porosity of Suslin sets. In particular, if A is a Suslin subset in a Banach space X, then each separable subspace of X can be enlarged to a separable subspace V such that A is sigma-porous in X if and only if the intersection of A and V is sigma-porous in V. Such a result is proved for several types of sigma-porosity. The proof is done using the method of elementary submodels, hence the results can be combined with other separable reduction theorems. As an application we extend a theorem of L.Zajicek on differentiability of Lipschitz functions on separable Asplund spaces to the nonseparable setting.