Triangle covering problems and the Viterbo inequality in the plane

TL;DR

This paper explores triangle covering problems and their connection to Viterbo's conjecture, providing new special cases, explanations of counterexamples, and proving the conjecture for certain Lagrangian products involving quadrilaterals.

Contribution

It establishes Viterbo's conjecture for Lagrangian products of convex shapes and quadrilaterals, offering new insights and explanations related to symplectic geometry and covering problems.

Findings

Proved Viterbo's conjecture for Lagrangian products with quadrilaterals.

Provided new special cases of Viterbo's conjecture.

Explained the counterexample of Haim-Kislev and Ostrover.

Abstract

We review a certain problem on covering triangles in the plane. Equivalently, it can be viewed as a family of 'isobilliard' inequalities in convex shapes, and as a special case of Viterbo's conjecture in symplectic geometry. We give an elementary overview of these topics and, using the optics of the covering problem, we establish several new special cases of Viterbo's conjecture, provide a simple explanation of the counterexample of Haim-Kislev and Ostrover, and state a few open questions. The main novel result is a proof of Viterbo's conjecture for lagrangian products $K \times Q$, where $Q \subset \mathbb{R}^2$ is any quadrilateral and $K \subset \mathbb{R}^2$ is any convex shape.