Zonoid theory and Hilbert's fourth problem (Q1112336)

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.