Applications of Hilbert's metric

Question

Among the fascinating constructions in mathematics is the Hilbert metric on a bounded convex subset of ${\mathbb R}^n$.

Where, within mathematics, is it used ? I know at least a proof of the Perron--Frobenius Theorem for non-negative matrices.

What are its applications in other sciences ?

Answer 1

Garrett Birkhoff used the Hilbert metric (he called it the projective metric) to prove that every $n$-by-$n$ matrix $A$ with positive entries is a Hilbert metric contraction on the cone of nonnegative vectors. He gave a formula for the contraction constant which is $\frac{1}{4} \arctan \Delta $ where $\Delta$ is the maximum Hilbert's (projective) metric distance between $Ae_i$ and $Ae_j$ where $e_i$ and $e_j$ are distinct elementary basis vectors. This immediately implies one aspect of the Perron-Frobenius theorem: that matrices with positive entries have a unique Perron eigenvector.

G. Birkhoff, Extensions of Jentzsch’s theorem. Trans. Amer. Math. Soc. 85 (1957), 219–227.

The Hilbert metric proof of the Perron-Frobenius theorem also extends to nonlinear maps which are monotone and homogeneous of degree one (i.e., $f(\lambda x) = \lambda f(x)$, $\forall \lambda > 0$). This is the primary advantage of using the Hilbert metric to prove the Perron-Frobenius theorem.

Answer 2

This might not be considered an application, but Hilbert metrics have been studied geometrically and dynamically. Here are several examples of questions that have been partially or totally answered:

• when is a convex domain endowed with its Hilbert metric $\delta$-hyperbolic?

• what is the volume entropy of Hilbert metrics?

• does there exist convex sets that admit a cocompact group of isometries (relative to their Hilbert metric)? (the answer is yes!)

I guess that amoung these, the last part can be considered an application: Hilbert metrics yields interesting subgroups of $\mathrm{PSL}(n,\mathbb{R})$.

For more details you can look at the works of Yves Benoist (in particular the "convexes divisibles" series and "Convexes hyperboliques et fonctions quasisymétriques", in french), Constantin Vernicos, Ludovic marquis and Mickaël Crampon.

Answer 3

Update

I found the following nice survey paper that lists lots of applications of Hilbert's metric (Birkhoff's version):

Birkhoff's version of Hilbert's metric and its applications in analysis by Bas Lemmens, Roger Nussbaum, April, 2013.

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Not Hilbert's metric directly, but immediately related is the Thompson metric which is frequently used.

I list below two interesting examples.

1. Solving nonlinear matrix equations:

   Invariant metrics, contractions and nonlinear matrix equations by H. Lee and Y. Lim

2. Dynamic Programming:

   Thompson metric, contraction property and differentiability of policy functions by L. Montrucchio

Answer 4

See e.g. pages 167-169 of "Topics in nonlinear analysis & applications" By Donald H. Hyers, George Isac, Themistocles M. Rassias. You can read it on Google books.