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1 [made Community Wiki]

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.