# On the volume of unit balls of finite-dimensional Lorentz spaces

## TL;DR

This paper investigates the volume of unit balls in finite-dimensional Lorentz spaces, providing formulas, asymptotic behavior, and applications to entropy numbers, advancing understanding of geometric properties of these function spaces.

## Contribution

It introduces explicit formulas and asymptotic estimates for the volume of unit balls in Lorentz spaces, including weak Lebesgue spaces, and applies these to entropy number decay analysis.

## Key findings

- Derived iterative formulas for volume in weak Lebesgue spaces
- Established asymptotic behavior of volume as dimension grows
- Characterized entropy number decay for specific embeddings

## Abstract

We study the volume of unit balls $B^n_{p,q}$ of finite-dimensional Lorentz sequence spaces $\ell_{p,q}^n.$ We give an iterative formula for ${\rm vol}(B^n_{p,q})$ for the weak Lebesgue spaces with $q=\infty$ and explicit formulas for $q=1$ and $q=\infty.$ We derive asymptotic results for the $n$-th root of ${\rm vol}(B^n_{p,q})$ and show that $[{\rm vol}(B^n_{p,q})]^{1/n}\approx n^{-1/p}$ for all $0<p<\infty$ and $0<q\le\infty.$ We study further the ratio between the volume of unit balls of weak Lebesgue spaces and the volume of unit balls of classical Lebesgue spaces. We conclude with an application of the volume estimates and characterize the decay of the entropy numbers of the embedding of the weak Lebesgue space $\ell_{1,\infty}^n$ into $\ell_1^n.$