Volumes of unit balls of mixed 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.