A Funk perspective on billiards, projective geometry and Mahler volume (Q6562506)

In this paper, the author proves several results on the Funk metric, a very natural but poorly studied non-symmetric metric defined on an arbitrary convex subset of a Euclidean space. In this sense, the study made in this paper is very welcome.\N\NThe first general remark the author makes is that although the Funk metric is not invariant under the projective transformations of the underlying manifold, it is projectively invariant up to the addition of an exact 1-form. This observation is at the heart of the several projective-invariant-like properties of the Funk metric that the author discovers. He shows in this paper that many of the metric invariants of the Funk metric are invariant under projective transformations and under projective duality. These invariants include the Holmes-Thompson volume and surface area of convex subsets and the length spectrum of their boundary. He studies Funk billiards, which generalize hyperbolic billiards in the same way that Minkowski billiards generalize Euclidean ones. He extends a result of Gutkin-Tabachnikov on the duality of Minkowski billiards. He next considers the volume of outward balls in Funk geometry. He conjectures a general affine inequality corresponding to the volume maximizers, which includes the Blaschke-Santalò and centro-affine isoperimetric inequalities as limiting cases and he proves this conjecture for unconditional bodies, yielding a new proof of the volume entropy conjecture for the Hilbert metric in this setting of unconditional bodies. He then deduces generalizations to higher moments of inequalities of Ball and Huang-Li, which in turn strengthen the Blaschke-Santalò inequality for unconditional bodies. Finally, he introduces a regularization of the total volume of a smooth strictly convex 2-dimensional set equipped with the Funk metric which is an analogue of the O'Hara Möbius energy of a knot, and he shows that it is a projective invariant of the convex body.\N\NThe results in this paper are collected into 6 theorems and several corollaries. These results constitutes a substantial contribution to Funk geometry. With these results, the author establishes new connections, based on the Funk metric, between projective geometry, billiards, convex geometry and some affine inequalities.