The Funk metric in the interior of a convex body is a lesser known relative of the Hilbert metric, which in turn generalizes the Beltrami-Klein model of hyperbolic geometry.
It recently became clear that the Funk geometry is related to various aspects of convex geometry, affine geometry, Finsler billiards, and the combinatorics of convex polyhedra.
Starting with the newly observed property of projective invariance of the Funk metric, I will present results relating Funk geometry to the Blaschke-Santalo inequality and Mahler's conjecture, the Colbois - Verovic conjecture (Tholozan's volume entropy theorem) in Hilbert geometry, Schaeffer's dual girth conjecture (Alvarez-Paiva's theorem) and Kalai's flag number conjecture for symmetric polyhedra. Partly based on joint work with C. Vernicos and C. Walsh.