formula for d coincides with the expression for the distance between points in the Cayley–Klein model of hyperbolic geometry, up to a multiplicative constant. If the cone K is the positive orthant in Rn then the induced metric on the projectivization of K is often called simply Hilbert's projective metric. This cone corresponds to a domain Ω which is a regular simplex of dimension n − 1. Motivation and applications Hilbert introduced his metric in order to construct an axiomatic metric geometry in which there exist triangles ABC whose vertices A, B, C are not collinear, yet one of the sides is equal to the sum of the other two — it follows that the shortest path connecting two points is not unique in this geometry. In particular, this happens when the convex set Ω is a Euclidean triangle and the straight line extensions of the segments AB, BC, AC do not meet the interior of one of the sides of Ω. Garrett Birkhoff used Hilbert's metric and the Banach contraction principle to rederive the Perron–Frobenius theorem in finite-dimensional linear algebra and its analogues for integral operators with positive kernels. Birkhoff's ideas have been further developed and used to establish various nonlinear generalizations of the Perron-Frobenius theorem, which have found significant uses in computer science, mathematical biology, game theory, dynamical systems theory, and ergodic theory. Generalizing earlier results of Anders Karlsson and Guennadi Noskov, Yves Benoist determined a system of necessary and sufficient conditions for a bounded convex domain in Rn, endowed with its Hilbert metric, to be a Gromov hyperbolic space. C. Vernicos and C. Walsh, then expanded upon by David Mount and Ahmed Abdelkader, showed that balls in the Hilbert Metric and Macbeath regions are approximately equivalent up to constant factors. References Metric geometry