Hilbert metric

In mathematics, the Hilbert metric, also known as the Hilbert projective metric, is an explicitly defined distance function on a bounded convex subset of the n-dimensional Euclidean space Rn. It was introduced by as a generalization of Cayley's formula for the distance in the Cayley–Klein model of hyperbolic geometry, where the convex set is the n-dimensional open unit ball. Hilbert's metric has been applied to Perron–Frobenius theory and to constructing Gromov hyperbolic spaces.

Definition
Let &Omega; be a convex open domain in a Euclidean space that does not contain a line. Given two distinct points A and B of &Omega;, let X and Y be the points at which the straight line AB intersects the boundary of &Omega;, where the order of the points is X, A, B, Y. Then the Hilbert distance d(A, B) is the logarithm of the cross-ratio of this quadruple of points:


 * $$ d(A,B)=\log\left(\frac{|YA|}{|YB|}\frac{|XB|}{|XA|}\right). $$

The function d is extended to all pairs of points by letting d(A, A) = 0 and defines a metric on &Omega;. If one of the points A and B lies on the boundary of &Omega; 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 closed convex cone K in a Banach space V (possibly, infinite-dimensional). In addition, the cone K is assumed to be pointed, i.e. K ∩ (&minus;K) = {0} and thus K determines a partial order $$\leq_K$$ on V. Given any vectors v and w in K \ {0}, one first defines


 * $$ M(v/w)=\inf\{\lambda:v\leq_K\lambda w\}, \quad m(v/w)=\sup\{\mu:\mu w \leq_K v\}. $$

The Hilbert pseudometric on K \ {0} is then defined by the formula


 * $$ d(v,w)=\log\frac{M(v/w)}{m(v/w)}. $$

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 &times; V is the cone over a convex set &Omega;,


 * $$ K=\{(t,tx): t\in\mathbb{R}, x\in\Omega \},$$

then the space of rays of K is canonically isomorphic to &Omega;. If v and w are vectors in rays in K corresponding to the points A, B ∈ &Omega; then these two formulas for d yield the same value of the distance.

Examples

 * In the case where the domain &Omega; 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 &Omega; which is a regular simplex of dimension n &minus; 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 &Omega; 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 &Omega;.
 * 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.