Hilbert geometry

The geometry of a complete metric space $H$ with a metric $h(x,y)$ which contains, together with two arbitrary, distinct points $x$ and $y$, also the points $z$ and $t$ for which $h(x,z)+h(z,y)=h(x,y)$, $h(x,y)+h(y,t)=h(x,t)$, and which is homeomorphic to a convex set in an $n$-dimensional affine space $A^n$, the geodesics $\gamma\in H$ being mapped to straight lines of $A^n$. Thus, let $K$ be a convex body in $A^n$ with boundary $\partial K$ not containing two non-collinear segments, and let $x,y\in K$ be located on a straight line $l$ which intersects $\partial K$ at $a$ and $b$; let $R(x,y,a,b)$ be the cross ratio of $x$, $y$, $a$, $b$ (so that if $x=(1-\lambda)a+\lambda b$, $y=(1-\mu)a+\mu b$, then $R(x,y,a,b)=\mu(1-\lambda)/\lambda(1-\mu)$). Then

$$h(x,y)=\frac12|\ln R(x,y,a,b)|$$

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

The problem of determining all metrizations of $K$ 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)

References

[a1]	H. Busemann, P.J. Kelly, "Projective geometry and projective metrics" , Acad. Press (1953)
[a2]	M. Berger, "Geometry" , I , Springer (1987)