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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 ?

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Excellent question Denis! I'd wanted to ask a question along these lines for quite some time now. –  Suvrit Sep 27 '11 at 9:08
How about a link or definition of the Hilbert distance? –  Gerald Edgar Sep 27 '11 at 14:48
Dear Denis, I would like to ask you, if you don't mind, to explain a bit more why you find this construction fascinating. Also, could you explain a bit how one can prove Perron-Frobenius theorem using Hilbert metric? Would this be some proof that is more conceptual, than a usual one? –  Dmitri Sep 27 '11 at 21:35
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4 Answers

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.

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This is precisely what I mentionned in my question. –  Denis Serre Sep 29 '11 at 5:43
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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.

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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.

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

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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.

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