arXiv subjects
Publications and source records attributed to Mark Berezovik.
Symplectically self-polar bodies have been introduced recently. They appeared in the context of Mahler's conjecture and in the context of symplectic outer billiard dynamics. In this paper we discuss the topology of the space of symplectically self-polar convex bodies. Namely, we show that this space is contractible.
We present a link between billiards in convex plane domains and Hofer's geometry, an area of symplectic topology. For smooth strictly convex billiard tables, we prove that the Hofer distance between the corresponding billiard ball maps admits an upper bound in terms of a simple geometric distance between the tables. We use this result to show that the billiard ball map of a convex polygon lies in the completion, with respect to Hofer's metric, of the group of smooth area-preserving maps of the annulus. Finally, we discuss related connections to dynamics and pose several open problems.
We state a conjecture about the volume of symplectically self-polar convex bodies and show that it is equivalent to Mahler's conjecture concerning the volume of a convex body and its Euclidean polar. We also establish lower and upper bounds for symplectic capacities of symplectically self-polar bodies.
It is known that $C^1$-smooth strictly convex Radon norms in $\mathbb{R}^2$ can be characterized by the property that the outer billiard map, which corresponds to the unit ball of the norm, has an invariant curve consisting of 4-periodic orbits. In higher dimensions, Radon norms are necessarily Euclidean. However, we show in this paper that the property of existence of an invariant curve of 4-periodic orbits allows a higher-dimensional extension to the class of symplectically self-polar convex bodies. Moreover, this class of convex bodies provides the first non-trivial examples of invariant hypersurfaces for outer billiard map. This is in contrast with conventional Birkhoff billiards in higher dimensions, where it was proved by Berger and Gruber that only ellipsoids have caustics. It is not known, however, if non-trivial invariant hypersurfaces can exist for higher-dimensional Birkhoff billiards.
In this paper we continue the study of symplectically self-polar convex bodies started in arXiv:2211.14630. We construct symplectically self-polar convex bodies of the minimal Ekeland-Hofer-Zehnder capacity. This in turn proves that the lower bound for the Ekeland-Hofer-Zehnder capacity for centrally symmetric convex bodies obtained in arXiv:1801.00242 cannot be improved. We also make some numerical experiments and speculations regarding the minimal volume of symplectically self-polar convex bodies.