Shortest Minkowski billiard trajectories on convex bodies

凸桌上的最短闭台球轨道

Shortest closed billiard orbits on convex tables

给定平面紧凸台球桌$T$，给出了求$T$上最短广义闭台球轨道的算法.（如果$T$具有光滑边界，则广义台球轨道是通常台球轨道。）如果$T$是多边形，则该算法是有限的，并且提供了一般的近似方案。作为一个例子，我们证明了正则$n$ -gon $R_n$中的最短广义闭台球轨道是$n \ge 4$的2-bounce，其长度是$R_n$宽度的两倍.作为应用，我们得到了在标准辛向量空间$\mathbb{R}^4$中计算四维域$T \times B^2$的Ekeland-Hohnder-Zehnder容量的算法.我们的方法是基于Bezdek-Bezdek的工作和Fagnano三角形在锐角三角形中的唯一性。它的作品，更一般地说，平面闵可夫斯基台球。

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。

Integrable Billiards on a Minkowski Hyperboloid: Extremal Polynomials and Topology

We consider billiard systems within compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We derive conditions for elliptic periodicity for such billiards. We describe the topology of those billiard systems in terms of Fomenko invariants. We provide then periodicity conditions in terms of functional Pell equations and related extremal polynomials. Several examples are computed in terms of elliptic functions and classical Chebyshev and Zolotarev polynomials, as extremal polynomials over one or two intervals. These results are contrasted with the cases …

Periodic trajectories of ellipsoidal billiards in the 3-dimensional Minkowski space

In this paper, we give detailed analysis and description of periodic trajectories of the billiard system within an ellipsoid in the 3-dimensional Minkowski space, taking into account all possibilities for the caustics. The conditions for periodicity are derived in algebro-geometric, analytic, and polynomial form.

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.

Periodic billiards within conics in the Minkowski plane and Akhiezer polynomials

We derive necessary and sufficient conditions for periodic and for elliptic periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining elliptic curve. We provide several examples of periodic and elliptic periodic trajectories with small periods. We observe relationship between Cayley-type conditions and discriminantly separable and factorizable polynomials. Equivalent conditions for periodicity and elliptic periodicity are derived in terms of polynomial-functional equations as well. The corresponding polynomials are related to the classical extremal …

An analogue of the Blaschke-Santaló inequality for billiard dynamics

The Blaschke-Santaló inequality is a classical inequality in convex geometry concerning the volume of a convex body and that of its dual. In this work we investigate an analogue of this inequality in the context of billiard dynamical system: we replace the volume with the length of the shortest closed billiard trajectory. We define a quantity called the "billiard product" of a convex body K, which is analogous to the volume product studied in the Blaschke-Santaló inequality. In the planar case, we derive an explicit expression for the billiard product in terms of the diameter of the body. We a…

Billiards on pythagorean triples and their Minkowski functions

It has long been known that the set of primitive pythagorean triples can be enumerated by descending certain ternary trees. We unify these treatments by considering hyperbolic billiard tables in the Poincare disk model. Our tables have m>=3 ideal vertices, and are subject to the restriction that reflections in the table walls are induced by matrices in the triangle group PSU^\pm_{1,1}\Zbb[i]. The resulting billiard map \tilde B acts on the de Sitter space x_1^2+x_2^2-x_3^2=1, and has a natural factor B on the unit circle, the pythagorean triples appearing as the B-preimages of fixed points. We…

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.

闭路空间上的周期台球轨线与莫尔斯理论

Periodic billiard trajectories and Morse theory on loop spaces

本文将莫尔斯理论应用于流形的闭路空间上的拉格朗日作用泛函，研究了紧致黎曼流形上的周期台球轨线。基于Benci-Giannoni的近似方法，证明了某对闭路空间的相对同调的非零性蕴含着周期台球轨线的存在性.我们还证明了道路空间的一个平行结果。我们应用这些结果来证明短台球轨迹和短测地线环路的存在。我们还恢复了关于凸体上最短周期台球轨线长度的两个已知结果：Ghomi不等式和Artstein-Ostrover的Brunn-Minkowski型不等式.

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…

关于凸体乘积边界上的哈密顿投影弹子球

On Hamiltonian projective billiards on boundaries of products of convex bodies

本文研究凸体乘积边界上的哈密顿投影弹子球，证明T-弹子球反射律为投影律当且仅当T是椭球，并将结果推广到Finsler弹子球，与辛等周猜想、Mahler猜想密切相关。