Zonoid theory and Hilbert's fourth problem
type: Item
instance of: scholarly article
title: Zonoid theory and Hilbert's fourth problem
zbMATH Open document ID: 0659.51022
DOI: 10.1007/BF00147451
author: Ralph Alexander
published in: Geometriae Dedicata
publication date: 1988-01-01T00:00:00Z
review text: A continuous metric d on \({\mathbb{R}}^ d\) is called projective if it is additive along lines. A special version of Hilbert's fourth problem asks for the determination of all projective metrics. A metric d is termed a hypermetric if \(\sum_{i<j}d(P_ i,P_ j)N_ iN_ j\leq 0\) for any points \(P_ 1,...,P_ m\) and any integers \(N_ 1,...,N_ m\) whose sum is 1. For a subset \(A\subset {\mathbb{R}}^ n\) let \(A\) denote the set of all hyperplanes of \({\mathbb{R}}^ n\) meeting A. The following theorem is the main result of the present paper: There is a natural linear isomorphism between the cone of projective hypermetrics d on \({\mathbb{R}}^ n\) and the cone of positive Borel measures \(\mu\) on the hyperplanesets of \({\mathbb{R}}^ n\) which for all points P, Q in \({\mathbb{R}}^ n\) satisfy the conditions \[ \mu P=0,\quad 0<\mu seg PQ<\infty. \] This isomorphism is determined by the generalized Crofton formula \(d(P,Q)=\mu seg PQ.\) This theorem generalizes a result of Pogorelov, (for \(n=2)\) and uses, as well as complements, work of \textit{Z. I. Szabó} [Adv. Math. 59, 185-301 (1986; Zbl 0608.53062)]. The paper emphasizes and uses the interesting connection with zonoids and generalized zonoids.
reviewed by: Rolf Schneider
Mathematics Subject Classification ID: 51K99
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 4078142
zbMATH Keywords: measure on hyperplanes
zbMATH Keywords: Hilbert's fourth problem
zbMATH Keywords: projective metrics
zbMATH Keywords: hypermetric
zbMATH Keywords: zonoids
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4078142