Bounds for Minkowski Billiard Trajectories in Convex Bodies

若Minkowski弹道是射影的，则它就是标准弹道

If a Minkowski billiard is projective, it is the standard billiard

本文通过简单证明展示，若Minkowski弹道同时为射影和Minkowski，则其为标准欧几里得弹道，且在C¹光滑条件下成立，并证明了半局部和局部版本的结果。

辛台球作为闵可夫斯基台球

Symplectic billiards as Minkowski billiards

研究建立闵可夫斯基台球与辛台球的联系，通过从典范辛结构约化描述闵可夫斯基台球映射，将辛台球视为其辛版本“平方根”，恢复并推广相关结果，如证明\(2n\)维中\(2r\)周期辛台球轨道的存在性。

A Brunn-Minkowski type inequality for extended symplectic capacities of convex domains and length estimate for a class of billiard trajectories

In this paper, we firstly generalize the Brunn-Minkowski type inequality for Ekeland-Hofer-Zehnder symplectic capacity of bounded convex domains established by Artstein-Avidan-Ostrover in 2008 to extended symplectic capacities of bounded convex domains constructed by authors based on a class of Hamiltonian non-periodic boundary value problems recently. Then we introduce a class of non-periodic billiards in convex domains, and for them we prove some corresponding results to those for periodic billiards in convex domains obtained by Artstein-Avidan-Ostrover in 2012.

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…

Shortest closed billiard orbits on convex tables

Given a planar compact convex billiard table $T$, we give an algorithm to find the shortest generalised closed billiard orbits on $T$. (Generalised billiard orbits are usual billiard orbits if $T$ has smooth boundary.) This algorithm is finite if $T$ is a polygon and provides an approximation scheme in general. As an illustration, we show that the shortest generalised closed billiard orbit in a regular $n$-gon $R_n$ is 2-bounce for $n \ge 4$, with length twice the width of $R_n$. As an application we obtain an algorithm computing the Ekeland-Hofer-Zehnder capacity of the four-dimensional domai…

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。