Symplectically self-polar polytopes of minimal capacity

TL;DR

This work advances the study of symplectically self-polar convex bodies by constructing a dimension-increasing family P^{⟂n} of such polytopes that attain the minimal EHZ capacity c_EHZ = 2 + 1/n and have explicit volume formulas, thereby confirming the sharpness of the known lower bounds. A new operation, the symplectic P-suspension, preserves self-polarity and yields precise volume scaling, enabling an inductive construction of minimal-capacity polytopes. The paper also conjectures that these polytopes minimize volume among all self-polar bodies and provides both 2D uniqueness results and extensive numerical evidence supporting minimal-volume behavior in higher dimensions, along with detailed appendices for auxiliary lemmas and enumerations.

Abstract

In this paper we continue the study of symplectically self-polar convex bodies started in arXiv:2211.14630. We construct symplectically self-polar convex bodies of the minimal Ekeland-Hofer-Zehnder capacity. This in turn proves that the lower bound for the Ekeland-Hofer-Zehnder capacity for centrally symmetric convex bodies obtained in arXiv:1801.00242 cannot be improved. We also make some numerical experiments and speculations regarding the minimal volume of symplectically self-polar convex bodies.

Figures (2)

• Figure 1: Hexagon $P$.
• Figure 2: Histograms of the volumes of the resulting polytopes for various $k$.

Theorems & Definitions (55)

• Theorem 1.1: Akopyan, Karasev, akopyan2017estimating
• Theorem 1.2
• Definition 2.1
• Remark 2.2
• Theorem 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• proof : Proof of Theorem \ref{['main:thm']}