Macroscopic dimension of the $\ell^p$-ball with respect to the $\ell^q$-norm

TL;DR

This paper investigates the large-scale geometric properties of the $\,\ell^p$-ball when measured using the $\,\ell^q$-norm, providing estimates of its macroscopic dimension.

Contribution

It offers new estimates for the macroscopic dimension of $\,\ell^p$-balls relative to the $\,\ell^q$-norm, advancing understanding of their large-scale geometry.

Findings

Derived bounds for the macroscopic dimension of $\,\ell^p$-balls

Provided estimates applicable across different $p$ and $q$ values

Enhanced understanding of the geometric structure of $\,\ell^p$-balls

Abstract

We show estimates of the "macroscopic dimension" of the $\ell^p$-ball with respect to the $\ell^q$-norm.