Volumes of unit balls of mixed sequence spaces

TL;DR

This paper derives a closed-form formula for the volume of unit balls in mixed norm sequence spaces, extending classical results and relevant for applications like group lasso in machine learning.

Contribution

It provides the first explicit formula for volumes of unit balls in mixed norm spaces, including real and complex cases, generalizing Dirichlet's classical results.

Findings

Derived a closed-form formula involving gamma functions

Extended classical volume calculations to mixed norm spaces

Applicable to real and complex sequence spaces

Abstract

The volume of the unit ball of the Lebesgue sequence space $\ell_p^m$ is very well known since the times of Dirichlet. We calculate the volume of the unit ball in the mixed norm $\ell^n_q(\ell_p^m)$, whose special cases are nowadays popular in machine learning under the name of group lasso. We consider the real as well as the complex case. The result is given by a closed formula involving the gamma function, only slightly more complicated than the one of Dirichlet. We close by an overview of open problems.