In mathematics, the **Hilbert metric**, also known as the **Hilbert projective metric**, is an explicitly defined distance function on a bounded convex subset of the *n*-dimensional Euclidean space**R**^{n}. It was introduced by David Hilbert (1895) as a generalization of Cayley’s formula for the distance in the Cayley–Klein model of hyperbolic geometry, where the convex set is the *n*-dimensional open unit ball. Hilbert’s metric has been applied to Perron–Frobenius theory and to constructing Gromov hyperbolic spaces.

## . . . Hilbert metric . . .

Let Ω be a convexopen domain in a Euclidean space that does not contain a line. Given two distinct points *A* and *B* of Ω, let *X* and *Y* be the points at which the straight line *AB* intersects the boundary of Ω, where the order of the points is *X*, *A*, *B*, *Y*. Then the **Hilbert distance***d*(*A*, *B*) is the logarithm of the cross-ratio of this quadruple of points:

The function *d* is extended to all pairs of points by letting *d*(*A*, *A*) = 0 and defines a metric on Ω. If one of the points *A* and *B* lies on the boundary of Ω then *d* can be formally defined to be +∞, corresponding to a limiting case of the above formula when one of the denominators is zero.

A variant of this construction arises for a closedconvex cone*K* in a Banach space*V* (possibly, infinite-dimensional). In addition, the cone *K* is assumed to be *pointed*, i.e. *K* ∩ (−*K*) = {0} and thus *K* determines a partial order

on *V*. Given any vectors *v* and *w* in *K* {0}, one first defines

The **Hilbert pseudometric** on *K* {0} is then defined by the formula

It is invariant under the rescaling of *v* and *w* by positive constants and so descends to a metric on the space of rays of *K*, which is interpreted as the projectivization of *K* (in order for *d* to be finite, one needs to restrict to the interior of *K*). Moreover, if *K* ⊂ **R** × *V* is the cone over a convex set Ω,

then the space of rays of *K* is canonically isomorphic to Ω. If *v* and *w* are vectors in rays in *K* corresponding to the points *A*, *B* ∈ Ω then these two formulas for *d* yield the same value of the distance.

## . . . Hilbert metric . . .

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