Abstract
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 a 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 also investigate upper bounds for this quantity in the class of polygons with a fixed number of vertices.
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Acknowledgments
This paper is part of the author’s Ph.D. thesis, being carried out under the joint supervision of Professor Misha Bialy and Professor Yaron Ostrover at Tel Aviv University. I would like to thank Professor Roman Karasev for his remarks. I would also like to thank Itai Bar-Deroma, Arnon Chor, Daniel Hadas, and Leonid Vishnevsky for their comments and discussions. The author is supported by ISF grants 580/20, 667/18, and by DFG grant MA-2565/7-1 within the Middle East Collaboration Program.
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Tsodikovich, D. An analogue of the Blaschke–Santaló inequality for billiard dynamics. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2634-9
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DOI: https://doi.org/10.1007/s11856-024-2634-9