Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Symplectic polarity and Mahler's conjecture

Mark Berezovik, Roman Karasev

TL;DR

The paper connects Mahler's conjecture to a symplectic-polarity framework by studying vol(X) for symplectically self-polar convex bodies X ⊂ $ℝ^{2n}$, proving a tight link between volume bounds and Mahler via X^ω = X. It establishes both lower and upper volume bounds, shows how sub-exponential factors can be suppressed to recover Mahler in full generality, and explores how symplectic reduction preserves self-polarity to relate high- and low-dimensional cases. The work also develops capacity-based constraints for self-polar bodies, deriving lower bounds on c_EHZ in terms of c_J and a robust lower bound for the affine cylindrical capacity c_ZA, along with sharpness results in dimension 2. Collectively, these results illuminate deep connections between convex geometry, symplectic topology, and classical conjectures, and provide structural tools (reductions, appendixed proofs) to tackle Mahler-type questions via symplectic methods.

Abstract

We state a conjecture about the volume of symplectically self-polar convex bodies and show that it is equivalent to Mahler's conjecture concerning the volume of a convex body and its Euclidean polar. We also establish lower and upper bounds for symplectic capacities of symplectically self-polar bodies.

Symplectic polarity and Mahler's conjecture

TL;DR

The paper connects Mahler's conjecture to a symplectic-polarity framework by studying vol(X) for symplectically self-polar convex bodies X ⊂ , proving a tight link between volume bounds and Mahler via X^ω = X. It establishes both lower and upper volume bounds, shows how sub-exponential factors can be suppressed to recover Mahler in full generality, and explores how symplectic reduction preserves self-polarity to relate high- and low-dimensional cases. The work also develops capacity-based constraints for self-polar bodies, deriving lower bounds on c_EHZ in terms of c_J and a robust lower bound for the affine cylindrical capacity c_ZA, along with sharpness results in dimension 2. Collectively, these results illuminate deep connections between convex geometry, symplectic topology, and classical conjectures, and provide structural tools (reductions, appendixed proofs) to tackle Mahler-type questions via symplectic methods.

Abstract

We state a conjecture about the volume of symplectically self-polar convex bodies and show that it is equivalent to Mahler's conjecture concerning the volume of a convex body and its Euclidean polar. We also establish lower and upper bounds for symplectic capacities of symplectically self-polar bodies.
Paper Structure (7 sections, 19 theorems, 72 equations)

This paper contains 7 sections, 19 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Sub-exponential factors in Mahler's inequality
  3. Proof of Theorem \ref{['theorem:mahler-polar']}
  4. Symplectic reduction of self-polar bodies
  5. Capacities of self-polar bodies
  6. Lower bounds on the affine cylindrical capacity
  7. Appendix

Key Result

Theorem 1.2

A lower bound of the form for $X\subset \mathbb R^{2n}$ with $X^\omega = X$ and a sub-exponential $c_n$, for all $n$, is equivalent to the validity of Mahler's conjecture for centrally symmetric bodies in all dimensions. Moreover, if this inequality is valid with a sub-exponential $c_n$, for all $n$, then it is valid in the

Theorems & Definitions (40)

  • Conjecture 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • ...and 30 more