Entropy numbers of spheres in Banach and quasi-Banach spaces

TL;DR

This paper establishes precise bounds on the entropy numbers of p-spheres in finite-dimensional Banach and quasi-Banach spaces, resolving previous gaps and extending to more general spaces.

Contribution

It provides sharp upper bounds for entropy numbers of p-spheres in $\, ext{ell}_q^d$ and generalizes results to quasi-Banach spaces, closing prior research gaps.

Findings

Sharp upper bounds for entropy numbers of p-spheres in $\, ext{ell}_q^d$

Resolution of a previously open problem in the field

Extension of bounds to quasi-Banach spaces

Abstract

We prove sharp upper bounds on the entropy numbers $e_k(S^{d-1}_p,\ell_q^d)$ of the $p$-sphere in $\ell_q^d$ in the case $k \geq d$ and $0< p \leq q \leq \infty$. In particular, we close a gap left open in recent work of the second author, T. Ullrich and J. Vybiral. We also investigate generalizations to spheres of general finite-dimensional quasi-Banach spaces.