Hilbert’s fourth problem

Abstract

Hilbert's fourth problem asks for the construction and the study of metrics on subsets of projective space for which the projective line segments are geodesics. Several solutions of the problem were given so far, depending on more precise interpretations of this problem, with various additional conditions satisfied. The most interesting solutions are probably those inspired by an integral formula that was first introduced in this theory by Herbert Busemann. Besides that, Busemann and his school made a thorough investigation of metrics defined on subsets of projective space for which the projective lines are geodesics and they obtained several results, characterizing several classes of such metrics. We review some of the developments and important results related to Hilbert's problem, especially those that arose from Busemann's work, mentioning recent results and connections with several branches of mathematics, including Riemannian geometry, the foundations of mathematics, the calculus of variations, metric geometry and Finsler geometry. Hilbert metrics – the subject of this handbook – constitute a basic class of metrics that satisfy the requirements of Hilbert's problem.