Hilbert metric

In the geometry of Hilbert metrics are certain metrics on bounded convex subsets of Euclidean space, which generalize the Beltrami -Klein model of hyperbolic geometry.

Definition

Let be a bounded, open, convex set. There is then a unique line through two distinct and intersections of these lines with the edge of any two points. The two intersections are denoted by, where closer and closer to lie on. Hilbert distance is then defined by the formula

For and.

The Hilbert metric is not always derived from a Riemannian metric, but always from a Finsler metric defined by

For.

Properties

Below are two compact convex sets and the two quantities associated with Hilbert metrics.

• It follows for all.
• If there is a linear map with, then for all.

Examples

• Be the unit sphere and the distance in the Beltrami -Klein model of hyperbolic space, then applies

Projective geometry

Be a true, open, convex subset of the projective space. ( A lot actually means, if there is an affine map containing, in the equivalent to a bounded set. ) One then defines the Hilbert metric on the Hilbert metric. Because the Hilbert metric is invariant under linear maps, This metric does not depend on the choice of the affine map.

Within the projective geometry can be interpreted as the cross-ratio of four points on the projective line through and certain.

The group of collineations

Is a Lie group and acts by isometries of the Hilbert metric, it can be isomorphic lift to a subgroup of.