Directional Lower Porosity

TL;DR

This paper explores the differences between upper and lower porosity in Banach spaces, revealing that lower porous sets can lack directional properties unlike upper porous sets, with specific counterexamples in the plane.

Contribution

It demonstrates that in finite-dimensional Banach spaces, upper porous sets are directionally upper porous, but lower porous sets can fail to be decomposed into countable unions of directionally lower porous sets.

Findings

Upper porous sets are directionally upper porous in finite-dimensional Banach spaces.

There exists a lower porous set in the plane not expressible as a countable union of directionally lower porous sets.

Lower porosity behaves fundamentally differently from upper porosity in terms of directional properties.

Abstract

We investigate differences between upper and lower porosity. In finite dimensional Banach spaces every upper porous set is directionally upper porous. We show the situation is very different for lower porous sets; there exists a lower porous set in the plane which is not even a countable union of directionally lower porous sets.