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Symplectic variations of convex bodies and the mean width

Jonghyeon Ahn, Ely Kerman

TL;DR

This work identifies toric symmetry as a key structural feature for convex bodies in ${\mathbb R}^{2n}$ that are critical for mean width under symplectomorphisms. It proves that for toric convex bodies, the mean width has zero first variation along any smooth Hamiltonian flow, and that the identity is a local (and in the smooth strictly convex case isolated) minimum for linear symplectic variations. The results are complemented by explicit analyses for ellipsoids and the Lagrangian bidisk, including global descent phenomena via Ramos’ embeddings, and by situating these findings within refinements of Urysohn-type inequalities and connections to symplectic capacities. Collectively, they establish toric symmetry as a natural and robust mechanism for variational stability of mean width under symplectic actions, while highlighting both local rigidity and global counterexamples in richer symplectic settings.

Abstract

In this work, we study convex bodies in $\RR^{2n}$ with the property that their mean width cannot be infinitesimally decreased by symplectomorphisms. The common theme of our results is that toric symmetry is a preferred feature of convex bodies with this property.

Symplectic variations of convex bodies and the mean width

TL;DR

This work identifies toric symmetry as a key structural feature for convex bodies in that are critical for mean width under symplectomorphisms. It proves that for toric convex bodies, the mean width has zero first variation along any smooth Hamiltonian flow, and that the identity is a local (and in the smooth strictly convex case isolated) minimum for linear symplectic variations. The results are complemented by explicit analyses for ellipsoids and the Lagrangian bidisk, including global descent phenomena via Ramos’ embeddings, and by situating these findings within refinements of Urysohn-type inequalities and connections to symplectic capacities. Collectively, they establish toric symmetry as a natural and robust mechanism for variational stability of mean width under symplectic actions, while highlighting both local rigidity and global counterexamples in richer symplectic settings.

Abstract

In this work, we study convex bodies in with the property that their mean width cannot be infinitesimally decreased by symplectomorphisms. The common theme of our results is that toric symmetry is a preferred feature of convex bodies with this property.
Paper Structure (19 sections, 25 theorems, 176 equations, 2 figures)

This paper contains 19 sections, 25 theorems, 176 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Motivation 1: Best positions in convex geometry
  4. Motivation 2: Symplectic refinements of Urysohn's inequality
  5. Motivation 3: Inequalities between the mean width and symplectic capacities
  6. Organization
  7. Proof of Theorem \ref{['variation']}
  8. Simplifications
  9. Coordinate expressions
  10. Proof of Proposition \ref{['form']}
  11. Proof of Theorem \ref{['linear']}
  12. Reducing to the case of a convex body $K$ with smooth and strictly convex boundary
  13. On the support function of $K$
  14. Restricting to symmetric positive definite symplectic matrices.
  15. Proof of Proposition \ref{['prop:lin']}
  16. ...and 4 more sections

Key Result

Theorem 1.3

If $K$ is a toric convex domain, then for any smooth path $\phi^t$ in $\operatorname{Symp}$ with $\phi^0 = \mathrm{Id}$, $t=0$ is a critical point of the function In fact, the derivative $\left.\frac{d}{dt}\right|_{t=0} M(\phi^t(K))$ exists and is equal to zero.

Figures (2)

  • Figure 1: $\mathrm{M}_{{\operatorname{Symp}}}(\mathbf{E}(a))^2/4$ lies below $c^B(\mathbf{E}(a))$ and strictly above $\sqrt{a}$, when there is room. For $1<a<2$, it also lies below ${{\mathrm {M}}}(\mathbf{E}(a))^2/4$.
  • Figure 2: Approximating $X_0 =\mu^{-1}(\Omega_0)$ by $X_1 =\mu^{-1}(\Omega_1)$

Theorems & Definitions (39)

  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Conjecture 1.6
  • Remark 1.7
  • Theorem 1.9: Green, gr
  • Theorem 1.10: Giannopoulos and Milman, gm
  • Proposition 1.11
  • Proposition 1.12
  • Proposition 1.14
  • ...and 29 more