Billiards in convex bodies with acute angles

On continuous billiard and quasigeodesic flows characterizing alcoves and isosceles tetrahedra

We characterize fundamental domains of affine reflection groups as those polyhedral convex bodies which support a continuous billiard dynamics. We interpret this characterization in the broader context of Alexandrov geometry and prove an analogous characterization for isosceles tetrahedra in terms of continuous quasigeodesic flows. Moreover, we show an optimal regularity result for convex bodies: the billiard dynamics is continuous if the boundary is of class $\mathcal{C}^{2,1}$. In particular, billiard trajectories converge to geodesics on the boundary in this case. Our proof of the latter co…

Elementary approach to closed billiard trajectories in asymmetric normed spaces

We apply the technique of Károly Bezdek and Daniel Bezdek to study billiard trajectories in convex bodies, when the length is measured with a (possibly asymmetric) norm. We prove a lower bound for the length of the shortest closed billiard trajectory, related to the non-symmetric Mahler problem. With this technique we are able to give short and elementary proofs to some known results.

A Funk perspective on billiards, projective geometry and Mahler volume

We explore connections furnished by the Funk metric, a relative of the Hilbert metric, between projective geometry, billiards, convex geometry and affine inequalities. We first show that many metric invariants of the Funk metric are invariant under projective transformations as well as projective duality. These include the Holmes-Thompson volume and surface area of convex subsets, and the length spectrum of their boundary, extending results of Holmes-Thompson and Álvarez Paiva on Schäffer's dual girth conjecture. We explore in particular Funk billiards, which generalize hyperbolic billiards in…

所有特征线均为平面的凸体

Convex bodies with all characteristics planar

该研究针对辛空间ℝ²ⁿ（n>1）中的光滑强凸体，证明了满足所有特征线或所有外弹子球轨迹为平面这一条件的凸体，均为球的仿射辛映像。

辛自极凸体空间的可收缩性

Contractibility of space of symplectically self-polar convex bodies

本文研究与Mahler猜想、辛外弹球动力学相关的辛自极凸体空间的拓扑，证明该空间具有可收缩性。

Shortest billiard trajectories

In this paper we prove that any convex body of the d-dimensional Euclidean space (d>1) possesses at least one shortest generalized billiard trajectory moreover, any of its shortest generalized billiard trajectories is of period at most d+1. Actually, in the Euclidean plane we improve this theorem as follows. A disk-polygon with parameter r>0 is simply the intersection of finitely many (closed) circular disks of radii r, called generating disks, having some interior point in common in the Euclidean plane. Also, we say that a disk-polygon with parameter r>0 is a fat disk-polygon if the pairwise …