Let Ω be a convexopen domain in a Euclidean space that does not contain a line. Given two distinct points A and B of Ω, let X and Y be the points at which the straight line AB intersects the boundary of Ω, where the order of the points is X, A, B, Y. Then the Hilbert distanced(A, B) is the logarithm of the cross-ratio of this quadruple of points:
The function d is extended to all pairs of points by letting d(A, A) = 0 and defines a metric on Ω. If one of the points A and B lies on the boundary of Ω then d can be formally defined to be +∞, corresponding to a limiting case of the above formula
when one of the denominators is zero.
A variant of this construction arises for a closedconvex coneK in a Banach spaceV (possibly, infinite-dimensional). In addition, the cone K is assumed to be pointed, i.e. K ∩ (−K) = {0} and thus K determines a partial order on V. Given any vectors v and w in K \ {0}, one first defines
The Hilbert pseudometric on K \ {0} is then defined by the formula
It is invariant under the rescaling of v and w by positive constants and so descends to a metric on the space of rays of K, which is interpreted as the projectivization of K (in order for d to be finite, one needs to restrict to the interior of K). Moreover, if K ⊂ R × V is the cone over a convex set Ω,
then the space of rays of K is canonically isomorphic to Ω. If v and w are vectors in rays in K corresponding to the points A, B ∈ Ω then these two formulas for d yield the same value of the distance.
Examples
In the case where the domain Ω is a unit ball in Rn, the 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 showed that balls in the Hilbert Metric and asymptotic balls are approximately equivalent up to constant factors.
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.
Nielsen, Frank; Sun, Ke (2017), "Clustering in Hilbert's Projective Geometry: The Case Studies of the Probability Simplex and the Elliptope of Correlation Matrices", Geometric Structures of Information, Signals and Communication Technology, pp. 297–331, arXiv:1704.00454, doi:10.1007/978-3-030-02520-5_11, ISBN978-3-030-02519-9, S2CID125430592
Nielsen, Frank; Shao, Laëtitia (2017), On Balls in a Hilbert Polygonal Geometry, vol. 77, LIPIcs-Leibniz International Proceedings in Informatics (SoCG), archived from the original on 2021-12-20
Papadopoulos, Athanase; Troyanov, Marc (2014). Handbook of Hilbert Geometry. European Mathematical Society.
Lemmens, Bas; Nussbaum, Roger (2012). Nonlinear Perron-Frobenius Theory. Cambridge Tracts in Mathematics. Vol. 189. Cambridge University Press.
Vernicos, Constantin; Walsh, Cormac (25 September 2018), "Flag-approximability of convex bodies and volume growth of Hilbert geometries", arXiv:1809.09471v1 [math.MG]
Abdelkader, Ahmed; Mount, David M. (2018), "Economical Delone Sets for Approximating Convex Bodies", 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018), 101: 4:1–4:12, doi:10.4230/LIPIcs.SWAT.2018.4