A regularity result for shortest generalized billiard trajectories in convex bodies in $\mathbb{R}^n$

从两点看不出镜面的物体

Bodies with mirror surface invisible from two points

在这里，我们关注的是台球不可见性的一个特殊问题，其中在欧几里得空间中具有分段光滑边界的有界集被等同于具有镜面的物体，并且在几何光学的框架中，集合的补中的台球被等同于物体外部光线的动力学。我们表明，在这种情况下，它是可以构建一个机构看不见的两点。

光滑凸体中的周期台球轨迹

Periodic billiard trajectories in smooth convex bodies

我们考虑台球轨迹在光滑凸体$\mathbb R^d$和估计的数量不同的周期性轨迹，使每周期的$p$反射在身体的边界。在素数$p$的情况下，我们得到了下界$(d-2)(p-1)+2$，这比以前的估计好得多。

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。

刚体碰撞和非标准台球的Differential Geometry

Differential Geometry of Rigid Bodies Collisions and Non-standard Billiards

由$\mathbb{R}^n$中的两个无约束刚体组成的机械系统的位形流形$M$，$n\geq 1$，是一个有边界带边流形（通常有奇点）的流形。完整的系统描述需要边界条件，这些条件规定了碰撞后轨道应如何继续。边界条件是在$M$的边界上的每个切空间处的碰撞映射的分配，其给出作为碰撞前状态的函数的系统的碰撞后状态。我们的主要结果是一个完整的描述空间的线性碰撞地图满足能量和（线和角）动量守恒，时间可逆性，和自然的要求，脉冲力只作用在切触点的碰撞机构。这些假设可以用几何语言来表述，即明确给出$M$边界切丛的一族向量子丛：对角子丛、非滑移子丛和脉冲子丛。在边界构型上的碰撞映射被证明是限制在非滑动子空间上的恒等式的等距对合。这种映射的空间自然地被识别为$\mathbb{R}^{n-1}$、$0\leq k\leq n-1$的$k$维子空间的格拉斯曼的联合，每个子空间指定接触粗糙度的方向。然后，我们考虑非标准的台球系统，通过固定的机构之一的位置，并允许不同的镜面反射的边界条件定义。我们还提出了一些意见的动力学性质的简单例子的非标准台球，并提供了一个充分条件的台球映射的边界状态的空间，以保持规范（刘维）措施的常能超曲面。

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…

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…

$C^1$-generic中心对称严格凸外弹子球中的无界轨道

Unbounded orbits in $C^1$-generic centrally symmetric strictly convex outer billiards

本文证明在$C^1$-generic中心对称严格凸外弹子球中，存在无界轨道，肯定回答了Moser和Neumann的著名问题，关键利用刚性定理和Mather扩散机制。

The Minkowski Billiard Characterization of the EHZ-capacity of Convex Lagrangian Products

We rigorously state the connection between the EHZ-capacity of convex Lagrangian products $K\times T\subset\mathbb{R}^n\times\mathbb{R}^n$ and the minimal length of closed $(K,T)$-Minkowski billiard trajectories. This connection was made explicit for the first time by Artstein-Avidan and Ostrover under the assumption of smoothness and strict convexity of both $K$ and $T$. We prove this connection in its full generality, i.e., without requiring any conditions on the convex bodies $K$ and $T$. This prepares the computation of the EHZ-capacity of convex Lagrangian products of two convex polytopes…

粗糙集上的台球散射：二维情形

Billiard scattering on rough sets: two-dimensional case

粗糙的二维（凸）体的概念被引入，每个粗糙的身体有分配一个措施上的$\TTT^3$描述台球散射的身体。主要结果是粗糙体产生的措施集的特征。这一结果可用于求解各种最小气动阻力问题。

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.

台球与二维最优阻力问题

Billiards and two-dimensional problems of optimal resistance

一个物体在由不相互作用的点粒子组成的介质中运动;粒子与物体的相互作用是绝对弹性的。需要找到使介质对其运动的阻力最小化或最大化的物体形状。这是最优阻力问题的一般设置，可以追溯到牛顿。在这里，我们限制自己的二维问题的旋转（一般非凸）机构。本文的主要结果如下。首先，对任意具有分段光滑边界$B \subset \mathbb{R}^2$的紧连通集，我们在$\mathbb{R}^2 \setminus B$中的台球生成的$\partial(\text{conv}B) \times [-π/2, π/2]$上赋予一个测度$ν_B$，并刻画了测度集$\{ν_B \}$ .其次，使用这个特征，我们解决了各种问题的最小和最大阻力的旋转机构，将它们减少到特殊的蒙格-康托洛维奇问题。

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…

Lens Rigidity in Scattering by Unions of Strictly Convex Bodies in $\R^2$

It was proved in \cite{NS1} that obstacles $K$ in $\R^d$ that are finite disjoint unions of strictly convex domains with $C^3$ boundaries are uniquely determined by the travelling times of billiard trajectories in their exteriors and also by their so called scattering length spectra. However the case $d = 2$ is not properly covered in \cite{NS1}. In the present paper we give a separate different proof of the same result in the case $d = 2$.