2022/03/04 by Daniel Rudolf, Rudolf, Daniel · 1 citation
Mathematics · #Point processes and geometric inequalities #Mathematics and Applications #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2203.02294
We use the generalized Minkowski billiard characterization of the EHZ-capacity of Lagrangian products in order to reprove that the 4-dimensional Viterbo conjecture holds for the Lagrangian products (any triangle/parallelogram in ℝ2)×(any convex body in ℝ2) and extend this fact to the Lagrangian products (any trapezoid in ℝ2)×(any convex body in ℝ2). Based on this analysis, we classify equality cases of this version of Viterbo's conjecture and prove that most of them can be proven to be symplectomorphic to Euclidean balls. As a by-product, we prove sharp systolic Minkowski billiard / worm problem inequalities. Furthermore, we discuss the Lagrangian products (any convex quadrilateral in ℝ2)×(any convex body in ℝ2) for which we show that the truth of Viterbo's conjecture would follow from the positive solution of a challenging Euclidean covering problem. Finally, we show that the flow associated to equality cases of Viterbo's conjecture for Lagrangian products in ℝ4--which turn out to be convex polytopes--is not Zoll in general, but that a weaker Zoll property, namely, that every characteristic almost everywhere away from lower-dimensional faces is closed and action-minimizing, does apply.