Surfaces Meeting Porous Sets in Positive Measure

TL;DR

This paper constructs a special porous set in high-dimensional Banach spaces where many smooth surfaces intersect it in positive measure, impacting the understanding of null sets related to Lipschitz function differentiability.

Contribution

It demonstrates the existence of a directionally porous set with non-meager intersections with C^1 surfaces in Banach spaces of dimension greater than n.

Findings

Existence of a directionally porous set with large intersection properties.

Decomposition of Banach spaces into porous and null sets relative to C^1 surfaces.

Implications for studying null sets in Lipschitz differentiability theory.

Abstract

Let n>2 and X be a Banach space of dimension strictly greater than n. We show there exists a directionally porous set P in X for which the set of C^1 surfaces of dimension n meeting P in positive measure is not meager. If X is separable this leads to a decomposition of X into a countable union of directionally porous sets and a set which is null on residually many C^1 surfaces of dimension n. This is of interest in the study of certain classes of null sets used to investigate differentiability of Lipschitz functions on Banach spaces.