Radon transformations and zonoids
type: Item
instance of: scholarly article
title: Radon transformations and zonoids
zbMATH Open document ID: 0896.52011
DOI: 10.1007/BF02310957
author: A. B. Sekerin
published in: Mathematical Notes
publication date: 1997-07-14T00:00:00Z
review text: The function \(H\) on \(\mathbb{R}^n\) is the support function of a zonoid if it is of the form \[ H(x)= \int_{S^{n-1}} \bigl| \langle x,u \rangle\bigr | d\mu(u) \] with a finite positive measure \(\mu\) on the unit sphere \(S^{n-1}\). The following theorem is proved. Let \(H\) be a subharmonic function on \(\mathbb{R}^n\) which is positively homogeneous of degree one (in particular, \(H\) can be a support function of a convex body). Then \(H\) coincides, up to a linear function, with the support function of some zonoid if and only if \[ \int_{\mathbb{R}^n} H(x)\Delta \varphi (x)dx\geq 0 \] for any \(C^\infty\) function \(\varphi\) on \(\mathbb{R}^n\) with nonnegative Radon transform.
reviewed by: Rolf Schneider
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 44A12
zbMATH DE Number: 1028502
zbMATH Keywords: zonoid
zbMATH Keywords: Radon transform
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1028502