On the volume of unit balls of finite-dimensional Lorentz spaces
Anna Doležalová, Jan Vybíral
TL;DR
The paper investigates the volumes of unit balls $B^n_{p,q}$ in finite-dimensional Lorentz spaces $\ell_{p,q}^n$, deriving recursive and explicit formulas for the weak Lebesgue case ($q=\infty$) and a closed-form expression for $q=1$. It establishes sharp asymptotics $[\operatorname{vol}(B^n_{p,q})]^{1/n} \asymp n^{-1/p}$ for all $0<p<\infty$ and $0<q\le\infty$, with $p=\infty$ treated separately, and studies how weak-Lebesgue volumes compare to classical Lebesgue volumes via an exponential-in-$n$ ratio for $p\le 2$. The paper then derives entropy-number decay rates for embeddings between Lorentz spaces, notably $e_k(id:\ell^n_{1,\infty}\to\ell^n_1)\approx \log(1+n/k)$ for $1\le k\le n$ and $e_k\approx 2^{-(k-1)/n}$ for $k\ge n$, linking volume estimates to embedding theory and approximation properties in finite dimensions.
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 $\ell_1^n.$