Shortest billiard trajectories |
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Authors: | Dániel Bezdek Károly Bezdek |
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Institution: | (1) Haskayne School of Business, University of Calgary, 2500 University Drive N.W., Calgary, AB, T2N 1N4, Canada;(2) Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary, AB, T2N 1N4, Canada |
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Abstract: | In this paper we prove that any convex body of the d-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 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 distances between the centers of its generating disks are at most r. 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 for ε-rounded disk-polygons obtained from fat disk-polygons by rounding them off using circular disks of radii ε > 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.
K. Bezdek partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant. |
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Keywords: | (fat) Disk-polygon (generalized) Billiard trajectory Shortest (generalized) billiard trajectory |
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