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 the 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 Ω be a convex open 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 d 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 = 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.
Hilbert balls in convex polygonal domains have .
A variant of this construction arises for a closed convex cone K in a Banach space V. In addition, the cone K is assumed to be pointed, i.e. K ∩ = and thus K determines a partial order on V. Given any vectors v and w in K \ , one first defines
The Hilbert pseudometric on K \ 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. 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