Shortest billiard trajectories
Shortest billiard trajectories
复制标题
最短台球轨迹
DOI：
10.1007/s10711-009-9353-6
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发表时间：
2009
影响因子：
0.5
通讯作者：
Karoly Bezdek
中科院分区：
文献类型：
--
作者：
D. Bezdek;Karoly Bezdek
本文证明了d维欧氏空间(d≥2)的任意凸体都具有至少一条最短广义台球轨迹，并且其任意最短广义台球轨迹的周期至多为d+1。实际上，在欧氏平面上我们对这个定理进行了如下改进。参数 r > 0 的圆盘多边形只是有限多个（闭合的）半径圆盘的交集，称为生成圆盘，在欧几里得平面中具有一些共同的内点。另外，如果其生成圆盘的中心之间的成对距离最多为，则参数 > 0 的圆盘多边形是胖圆盘多边形。我们证明任意胖圆盘多边形的任何最短广义台球轨迹都是二周期轨迹。此外，我们还证明了通过使用半径ε > 0 的圆盘对胖盘多边形进行四舍五入而从胖盘多边形获得的 ε 圆盘多边形的模拟结果。我们的定理对 S. Zelditch 提出的关于表征凸体（其最短周期台球轨迹为周期 2）的最近问题给出了部分答案。
In this paper we prove that any convex body of thed-dimensional Euclidean space (d≥ 2) possesses at least one shortest generalized billiard trajectory moreover, any of its shortest generalized billiard trajectories is of period at mostd+ 1. Actually, in the Euclidean plane we improve this theorem as follows. A disk-polygon with parameterr> 0 is simply the intersection of finitely many (closed) circular disks of radiir, called generating disks, having some interior point in common in the Euclidean plane. Also, we say that a disk-polygon with parameterr> 0 is a fat disk-polygon if the pairwise distances between the centers of its generating disks are at mostr. We prove that any of the shortest generalized billiard trajectories of an arbitrary fat disk-polygon is a 2-periodic one. Also, we give a proof of the analogue result forε-rounded disk-polygons obtained from fat disk-polygons by rounding them off using circular disks of radiiε> 0. Our theorems give partial answers to the very recent question raised by S. Zelditch on characterizing convex bodies whose shortest periodic billiard trajectories are of period 2.