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

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