On the symplectic capacity and mean width of convex bodies
Jonghyeon Ahn, Ely Kerman
TL;DR
The paper investigates how the symplectic capacity $c_{EH}$ of a convex body $K \subset {\mathbb R}^{2n}$ relates to its mean width $M(K)$, highlighting both the known bound $c_{EH}(K) \le \frac{M(K)^2}{4}$ and refined formulations via symplectomorphisms and quermassintegrals. It presents an alternative derivation of this inequality from the symplectic Brunn–Minkowski framework, connecting to the local Viterbo conjecture of Abbondandolo–Benedetti and using Steiner's formula to bridge capacities with quermassintegrals. The authors then explore toric symmetry as a potential criterion for mean width minimization under symplectic images, proposing that $M_{Symp}(K)=M(K)$ holds precisely when $QK$ is toric for some $Q \in U(n)$, and establish a local minimality/stability picture with respect to toric configurations. A substantial part of the work identifies a broad family of linear-minimizing bodies for which the identity is an isolated local minimum, yet constructs non-toric counterexamples and demonstrates that certain nonlinear symplectomorphisms can decrease the mean width, thereby testing and refining the conjectural picture. Overall, the results sharpen the understanding of how symplectic transformations interact with classical convex-geometric measurements and point to a nuanced role for toric symmetry in guiding mean-width optimization.
Abstract
In this note we consider two topics involving the relationship between the symplectic capacity and the mean width of convex bodies in $\mathbb{R}^{2n}$. We first describe an alternative path from the symplectic Brunn-Minkowski inequality of Artstein-Avidan and Ostrover to another inequality, established by the same authors, that relates the capacity and mean width of convex bodies. This new path is less direct but it relates these inequalities to the quermassintegrals of convex bodies and to the local version of Viterbo's conjecture established by Abbondandolo and Benedetti for domains sufficiently close to the ball. We then consider the problem of identifying convex bodies whose mean width cannot be decreased by natural classes of symplectomorphisms. We state a conjectured characterization of convex bodies whose mean width is already minimal among all their symplectic images. To test this conjecture we identify a simple class of quadratic convex bodies whose mean width can not be decreased by linear symplectic maps near the identity. We then identify a subset of these examples that fail to satisfy the toric conditions of the conjecture, and show that one can find a nonlinear symplectomorphism that decreases their mean width.