Volumes of unit balls of mixed sequence spaces. Issue 8 (20th September 2016)

• Record Type: Journal Article
• Authors: Kempka, Henning; Vybíral, Jan
• Is Part Of: Mathematische Nachrichten. Volume 290:Issue 8/9(2017)
• Page Start: 1317
• Page End: 1327
• Publication Date: 2016-09-20
• Subjects: Mixed sequence spaces -- volumes of convex bodies -- Gaussian integrals -- 26B15 -- 46A45; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201500414
• Languages: English
• ISSNs: 0025-584X

Abstract

Abstract : The volume of the unit ball of the Lebesgue sequence space ℓ p m is very well known since the times of Dirichlet. We calculate the volume of the unit ball of the mixed norm ℓ q n ( ℓ p m ), whose special cases are nowadays popular in machine learning under the name of group Lasso. We give two proofs of the main results, one in the spirit of Dirichlet, the other one using polarization identities. The result is given by a closed formula involving the gamma function, only slightly more complicated than the one of Dirichlet. We consider the real as well as the complex case. We also consider the anisotropic unit balls. We close by an overview of open problems.