Hilbert's fourth problem. I
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 (1978; Zbl 0379.50002)]. The contents of the present paper are best described by an excerpt from the author's introduction: ''In Section 2 we sketch Ambartzumian's proof with some modification and prove a theorem which asserts that this 2- dimensional elementary method cannot be used directly in the higher- dimensional cases. In Section 3 we compute the density of \(\mu\) for the smooth Desarguesian plane metrics. In Section 4 we give a new elementary proof for the Pogorelov theorem in higher dimensions. The ideas of this proof are close to Ambartzumian's ideas given for the 2-dimensional case. In Section 5 we construct a Desarguesian metric which cannot be constructed by the B-B construction. In Sections 6 and 7 a second new proof is given for the Pogorelov theorem for general n-dimensional cases. The theorem is restated in the following generalized form: Any n- dimensional Desarguesian space of class \(C^{n+2}\), \(n>2\), can be constructed by the B-B construction. From the proof it seems clear that for the construction of general continuous Desarguesian metrics we need to use distributions. This distribution L, constructed in Section 8, is not a ''usual'' one, as it is impossible to define a metric from a general distribution in a natural manner. From our distribution L, also the metric can be defined in a natural manner. Summarized, the construction by the distribution L is the most general one by which we get all Desarguesian metrics.'' In a sense, this impressive work, which uses a variety of geometric and analytic ideas, can be considered as the first entirely general and comprehensive treatment of Hilbert's fourth problem. In Section 9, the problem is approached from another point of view, and it is shown how Desarguesian spaces can be considered as generalized Minkowski spaces. Still another construction for Desarguesian metrics, by means of the curvature of the space, is sketched in Section 10 and will be carried out in the second part of this paper. \{Reviewer's remark. Recently Alexander has proved (to appear) that a Desarguesian metric can be obtained by the B-B construction with a positive measure \(\mu\) if and only if it is a hypermetric. He utilizes connections of Hilbert's fourth problem with the theory of zonoids and also remarks that some parts of the author's work, in particular Section 5, can be shortened if known results on generalized zonoids, in part due to the reviewer [Math. Nachr. 44, 55-75 (1970; Zbl 0162.543)] are applied.\}
scientific article; zbMATH DE number 3984079
scientific article