---
tag: P923DHV74J
kind: concept
name: Hilbert metric
msc: [51F99, 52A20, 53C60]
specializes: [0VKD1TP7F3]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert_metric
same_as:
  wikidata: Q5761213
  wikipedia: Hilbert metric
---

## Summary

The Hilbert metric is a distance on any bounded open [[convex set|02AS45T3H9]] of $\R^n$.
It is the logarithm of a [[cross-ratio|09ASMRMD67]] of the two points and the two boundary points on the line through them.
It turns the convex set into a [[metric space|0VKD1TP7F3]] in which straight segments are shortest paths.
When the convex set is a ball, it is the [[Beltrami–Klein model|1WN9B18H86]] of hyperbolic space, so Hilbert geometries generalize the hyperbolic [[Cayley–Klein metric|J7ZE3EGRYX]].

## Explanations

### L1 human

Take a convex region, such as a disc or a triangle.
To measure the distance between two points, extend the segment between them until it hits the boundary on both sides.
The distance compares how close each point is to each boundary point, through a logarithm.
Near the boundary, distances blow up, so the boundary is infinitely far away.
In a disc this gives a picture of the hyperbolic plane in which lines look straight.