Hilbert metric

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

definition

A compact convex set.

Let be a bounded , open , convex set . For every two points there is then a clear straight line through and two clear intersection points of this straight line with the edge . The two points of intersection are denoted by, being closer to and closer to . The Hilbert distance is then defined by the formula ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle x,y\in \Omega }$ ${\displaystyle x,y}$ ${\displaystyle \partial \Omega }$ ${\displaystyle a,b}$ ${\displaystyle a}$ ${\displaystyle x}$ ${\displaystyle b}$ ${\displaystyle y}$ ${\displaystyle d_{H}}$ ${\displaystyle \Omega }$

${\displaystyle d_{Hilb}(x,y):=\log {\frac {\parallel y-a\parallel .\parallel x-b\parallel }{\parallel x-a\parallel .\parallel y-b\parallel }}}$

for and . ${\displaystyle x\not =y}$ ${\displaystyle d_{Hilb}(x,x)=0}$

The Hilbert metric does not always come from a Riemannian metric , but always from a Finsler metric defined by

${\displaystyle F(v_{x}):={\frac {d}{dt}}\mid _{t=0}d_{Hilb}(x,x+tv_{x})}$

for . ${\displaystyle x\in \Omega \subset \mathbb {R} ^{n},v_{x}\in T_{x}\Omega \cong \mathbb {R} ^{n}}$

properties

In the following we assume two compact, convex sets and the Hilbert metrics assigned to the two sets. ${\displaystyle \Omega _{1},\Omega _{2}\subset \mathbb {R} ^{n}}$ ${\displaystyle d_{1},d_{2}}$

• From follows for everyone . ${\displaystyle \Omega _{1}\subset \Omega _{2}}$ ${\displaystyle d_{1}(x,y)\geq d_{2}(x,y)}$ ${\displaystyle x,y\in \Omega _{1}}$
• If there is a linear mapping with , then it is for everyone . ${\displaystyle A:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle \Omega _{2}=A(\Omega _{1})}$ ${\displaystyle d_{1}(x,y)=d_{2}(Ax,Ay)}$ ${\displaystyle x,y\in \Omega _{1}}$

Examples

• Let be the unit sphere and the distance in the Beltrami-Klein model of hyperbolic space , then applies ${\displaystyle \Omega =\mathbb {D} ^{n}}$ ${\displaystyle d_{Hyp}}$

${\displaystyle d_{Hilb}=2d_{Hyp}}$.

Projective geometry

Be an actual, open, convex subset of projective space . (A set actually means if there is a containing affine map in which corresponds to a bounded set .) The Hilbert metric on is then defined by the Hilbert metric on . Because the Hilbert metric is invariant under linear maps, the metric defined in this way does not depend on the choice of the affine map. ${\displaystyle \Omega \subset \mathbb {R} P^{n}}$ ${\displaystyle \Omega \subset \mathbb {R} P^{n}}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega \subset U\cong V\subset \mathbb {R} ^{n}}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega ^{\prime }\subset \mathbb {R} ^{n}}$ ${\displaystyle \Omega \subset \mathbb {R} P^{n}}$ ${\displaystyle \Omega ^{\prime }\subset \mathbb {R} ^{n}}$

Within projective geometry, one can interpret it as the double ratio of the four points on the projective straight line determined by and . ${\displaystyle d_{Hilb}(x,y)}$ ${\displaystyle a,x,b,y}$ ${\displaystyle x}$ ${\displaystyle y}$

The group of collineations

${\displaystyle Coll(\Omega )=\left\{g\in PGL(n+1,\mathbb {R} ):g\Omega =\Omega \right\}}$

is a Lie group and works through isometrics of the Hilbert metric, it can be raised isomorphically to a subgroup of . ${\displaystyle SL(n+1,\mathbb {R} )}$

Applications

The Hilbert metric on is used in Birkhoff's proof of Perron-Fronenius' theorem . ${\displaystyle P(\mathbb {R} _{+}^{n})}$

Web links

• Images des Maths: Géométrie de Hilbert

literature

• Yves Benoist: A survey on divisible convex sets ( PDF; 165 kB )
• Ludovic Marquis: Around groups in Hilbert geometry ( PDF; 2.5 MB )