# Shortest closed billiard trajectories in the plane and equality cases in Mahler’s conjecture

@article{Balitskiy2014ShortestCB,
title={Shortest closed billiard trajectories in the plane and equality cases in Mahler’s conjecture},
author={Alexey Balitskiy},
journal={Geometriae Dedicata},
year={2014},
volume={184},
pages={121-134}
}
In this note we prove some Rogers–Shepard type inequalities for the lengths of shortest closed billiard trajectories, mostly in the planar case. We also establish some properties of closed billiard trajectories in Hanner polytopes, having some significance in the symplectic approach to the Mahler conjecture.
2 Citations

## Figures from this paper

Mahler’s conjecture for some hyperplane sections
We use symplectic techniques to obtain partial results on Mahler's conjecture about the product of the volume of a convex body and the volume of its polar. We confirm the conjecture for hyperplane
Equality cases in Viterbo's conjecture and isoperimetric billiard inequalities
In this note we apply the billiard technique to deduce some results on Viterbo's conjectured inequality between volume of a convex body and its symplectic capacity. We show that the product of a

## References

SHOWING 1-10 OF 12 REFERENCES
Elementary approach to closed billiard trajectories in asymmetric normed spaces
• Mathematics
• 2016
We apply the technique of K\'aroly 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
Bounds for Minkowski Billiard Trajectories in Convex Bodies
• Mathematics
• 2011
In this paper we use the Ekeland-Hofer-Zehnder symplectic capacity to provide several bounds and inequalities for the length of the shortest periodic billiard trajectory in a smooth convex body in
Shortest periodic billiard trajectories in convex bodies
AbstractWe show that the length of any periodic billiard trajectory in any convex body $$K \subset \mathbf{R}^n$$ is always at least 4 times the inradius of K; the equality holds precisely when
From symplectic measurements to the Mahler conjecture
• Mathematics
• 2014
In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains, and Mahler's conjecture on the
Elementary results in non-reflexive Finsler billiards
• Mathematics
• 2014
We apply the technique of K\'aroly Bezdek and Daniel Bezdek to study the billiards in convex bodies with non-reflexive Finsler/Minkowski norm, give elementary proofs of some known results and prove
Shortest billiard trajectories
• Mathematics
• 2009
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
Metric and isoperimetric problems in symplectic geometry
Introduction 411 1. Some basic results in symplectic topology 413 2. Capacity and symplectic reduction 414 3. Volume estimates for Lagrange submanifolds 416 3.1. The case of R. 417 3.2. Deformations
Contact geometry and isosystolic inequalities
• Mathematics
• 2011
A long-standing open problem asks whether a Riemannian metric on the real projective space with the same volume as the canonical metric carries a periodic geodesic whose length is at most π. A
Embedding Problems in Symplectic Geometry
The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large
Open question: the Mahler conjecture on convex bodies
• terrytao.wordpress.com/2007/03/08/open-problem-the-mahler-conjecture-on-convex-bodies/,
• 2007