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:

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 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 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 ∩ (−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 × V is the cone over a convex set Ω,

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

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

Motivation and applications

References

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