On Separable Determination of Sigma-P-Porous Sets in Banach Spaces

TL;DR

This paper develops a method using elementary submodels to reduce the analysis of sigma-P-porous sets in Banach spaces to separable subspaces, with applications to differentiability of approximately convex functions.

Contribution

It introduces a new approach involving elementary submodels for separable reduction of sigma-P-porous sets in Banach spaces, including Asplund spaces.

Findings

Separable reduction theorems for sigma-P-porous sets in Banach spaces.

Application to Frechet differentiability of approximately convex functions in Asplund spaces.

Extension of porosity concepts via elementary submodels.

Abstract

We use a method involving elementary submodels and a partial converse of Foran lemma to prove separable reduction theorems concerning Suslin sigma-P-porous sets where "P" can be from a rather wide class of porosity-like relations in complete metric spaces. In particular, we separably reduce the notion of Suslin cone small set in Asplund spaces. As an application we prove a theorem stating that a continuous approximately convex function on an Asplund space is Frechet differentiable up to a cone small set.