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arXiv · 1505.05867

Volumes of unit balls of mixed sequence spaces

Henning Kempka·Jan Vybíral
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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.

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Henning Kempka, Jan Vybíral. 2015-06-25. Volumes of unit balls of mixed sequence spaces. https://arxiv.org/abs/1505.05867

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