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 David Hilbert (1895) 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 Hilbert distance d(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.
