# Hilbert geometry

The geometry of a complete metric space with a metric which contains, together with two arbitrary, distinct points and , also the points and for which , , and which is homeomorphic to a convex set in an -dimensional affine space , the geodesics being mapped to straight lines of . Thus, let be a convex body in with boundary not containing two non-collinear segments, and let be located on a straight line which intersects at and ; let be the cross ratio of , , , (so that if , , then ). Then

is the metric of a Hilbert geometry (a Hilbert metric). If is centrally symmetric, then is a Minkowski metric (cf. Minkowski geometry); if is an ellipsoid, then defines the Lobachevskii geometry.

The problem of determining all metrizations of for which the geodesics are straight lines is Hilbert's fourth problem; it has been completely solved [4].

Geodesic geometry is a generalization of Hilbert geometry.

Hilbert geometry was first mentioned in 1894 by D. Hilbert in a letter to F. Klein.

#### References

[1] | D. Hilbert, "Grundlagen der Geometrie" , Springer (1913) |

[2] | "Hilbert problems" Bull. Amer. Math. Soc. , 8 (1902) pp. 437–479 (Translated from German) |

[3] | H. Busemann, "The geometry of geodesics" , Acad. Press (1955) |

[4] | A.V. Pogorelov, "Hilbert's fourth problem" , Winston & Wiley (1974) (In Russian) |

#### Comments

#### References

[a1] | H. Busemann, P.J. Kelly, "Projective geometry and projective metrics" , Acad. Press (1953) |

[a2] | M. Berger, "Geometry" , I , Springer (1987) |

**How to Cite This Entry:**

Hilbert geometry. M.I. Voitsekhovskii (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Hilbert_geometry&oldid=13680