Hilbert's fourth problem. I
type: Item
instance of: scholarly article
title: Hilbert's fourth problem. I
zbMATH Open document ID: 0608.53062
DOI: 10.1016/0001-8708(86)90056-3
published in: Advances in Mathematics
publication date: 1986-01-01T00:00:00Z
review text: After some reductions provided by the foundations of geometry and by Busemann's theory of G-spaces, Hilbert's fourth problem comes down to the following. Let D be a subset of the projective space \(P^ n\) which is either the whole space or an open convex subset of an affine subspace \(A^ n\) of \(P^ n\). The problem is to construct all metrics d on D which (1) are continuous and complete, (2) are additive on the lines, (3) satisfy the strict triangle inequality for non-collinear triples of points. A metric with these properties is called a Desarguesian metric. Let \(\mu\) be a signed Borel measure on the space of hyperplanes of \(P^ n\) meeting D, which satisfies the following properties: (1) The set of hyperplanes through any fixed point \(P\in D\) has \(\mu\)-measure zero. (2) The set of hyperplanes intersecting two non-collinear segments in D with a common endpoint has positive measure. (3) If \(D=P^ n\), then \(\mu\) is finite; if \(D=P^ n\), the set of hyperplanes meeting any fixed ray in D has infinite \(\mu\)-measure. For any such \(\mu\), let \(d_{\mu}(x,y)\) be the \(\mu\)-measure of the hyperplanes intersecting the segment with endpoints x,y\(\in D\). Then \(d_{\mu}\) is a Desarguesian metric on D; it is said to be obtained by the B-B construction. The following theorem is due to \textit{A. V. Pogorelov} [Hilbert's fourth problem (Moscow 1974; Zbl 0291.52008)]: (1) Any 2-dimensional Desarguesian metric can be obtained by the B-B construction (with a positive measure). (2) Any 3-dimensional Desarguesian metric of class \(C^{\infty}\) can be obtained by the B-B construction. Every 3-dimensional Desarguesian metric can be approximated, uniformly on each compact set, by Desarguesian metrics of class \(C^{\infty}\). More general versions of the 2-dimensional case, obtained by different methods, are due to \textit{R. V. Ambartzumian} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 37, 145-155 (1976; Zbl 0325.28020)] and \textit{R. Alexander} [Ill. J. Math. 22, 177-190 …
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 53C60
Mathematics Subject Classification ID: 51K99
Mathematics Subject Classification ID: 53C70
zbMATH DE Number: 3984079
zbMATH Keywords: projective metric
zbMATH Keywords: measure on hyperplanes
zbMATH Keywords: G-spaces
zbMATH Keywords: Hilbert's fourth problem
zbMATH Keywords: Desarguesian metric
zbMATH Keywords: B-B construction
zbMATH Keywords: distributions
zbMATH Keywords: generalized Minkowski spaces
author: Zoltán Imre Szabó
reviewed by: Rolf Schneider
MaRDI profile type: Publication
full work available at URL: 0001-8708(86)90056-3
OpenAlex ID: W2038785898
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description: scientific article; zbMATH DE number 3984079