Hilbert metric

Summary

The Hilbert metric is a distance on any bounded open convex set of $\R^n$. It is the logarithm of a cross-ratio of the two points and the two boundary points on the line through them. It turns the convex set into a metric space in which straight segments are shortest paths. When the convex set is a ball, it is the Beltrami–Klein model of hyperbolic space, so Hilbert geometries generalize the hyperbolic Cayley–Klein metric.