[Paper Review] A Counterexample to Viterbo's Conjecture

Pazit Haim‐Kislev, Yaron Ostrover|arXiv (Cornell University)|May 26, 2024

Mathematics and Applications4 citations

TL;DR

This paper presents a counterexample to Viterbo's volume-capacity conjecture for all dimensions $ n \geq 2 $, using Minkowski billiard dynamics to compute the Ekeland–Hofer–Zehnder capacity of a Lagrangian product of two rotated regular pentagons in $ \mathbb{R}^4 $. The key result shows that the capacity-to-volume ratio exceeds the Euclidean ball's value, proving that symplectic capacities are not universally bounded by volume in the convex domain class.

ABSTRACT

We present a counterexample to Viterbo's volume-capacity conjecture. This implies, in particular, that in contrast with a well-known conjecture, symplectic capacities do not coincide on the class of convex domains in the classical phase space.

Motivation & Objective

• To disprove Viterbo’s long-standing volume-capacity conjecture, which posits that the Euclidean ball maximizes symplectic capacity among convex domains of fixed volume.
• To investigate whether symplectic capacities are universally bounded by volume in the class of convex domains in $ \mathbb{R}^{2n} $.
• To demonstrate that the Ekeland–Hofer–Zehnder capacity of Lagrangian products of convex bodies can be computed via Minkowski billiard dynamics.
• To construct a concrete counterexample in $ \mathbb{R}^4 $ using symmetric convex bodies—specifically, a regular pentagon and its 90° rotation—showing violation of the conjectured inequality.

Proposed method

• The authors use the Ekeland–Hofer–Zehnder capacity, defined as the minimal action of closed characteristics on the boundary of a convex domain.
• They compute the capacity of the Lagrangian product $ K \times T $, where $ K $ is a regular pentagon and $ T $ is its 90° rotation in $ \mathbb{R}^4 $, via Minkowski billiard trajectories.
• The capacity is shown to equal the $ T^\circ $-length of a 2-bounce $ T $-billiard trajectory along a diagonal of $ K $, leveraging duality and symmetry in the billiard dynamics.
• The $ T^\circ $-length is computed explicitly using support functionals: $ \|x\|_{T^\circ} = \langle x, w_i \rangle $, where $ w_i $ are vertices of the dual body.
• The proof relies on minimizing the $ T^\circ $-length over 3-vertex billiard paths under geometric constraints (non-translatability, direction constraints), showing the minimum occurs at a 2-bounce trajectory.
• A continuity and approximation argument is used to extend the counterexample from a non-smooth pentagon to a smooth convex body, preserving the violation of Viterbo’s inequality.

Experimental results

Research questions

• RQ1 Does Viterbo’s volume-capacity conjecture hold for all convex domains in $ \mathbb{R}^{2n} $, or are there counterexamples?
• RQ2 Can symplectic capacities differ on the class of convex domains, even when volume is fixed?
• RQ3 Is the Ekeland–Hofer–Zehnder capacity of a Lagrangian product of convex bodies computable via Minkowski billiard dynamics?
• RQ4 Can a non-spherical convex domain achieve a higher symplectic capacity-to-volume ratio than the Euclidean ball?

Key findings

• The Ekeland–Hofer–Zehnder capacity of the Lagrangian product $ K \times T $, where $ K $ is a regular pentagon and $ T $ its 90° rotation, is $ c_{\text{EHZ}}(K \times T) = 2\cos(\pi/10)(1 + \cos(\pi/5)) $.
• The capacity-to-volume ratio satisfies $ \frac{c_{\text{EHZ}}(K \times T)^2}{2A^2} = \frac{\sqrt{5}+3}{5} > 1 $, where $ A = \frac{5}{2}\sin(2\pi/5) $ is the area of $ K $ or $ T $, violating Viterbo’s conjecture.
• The counterexample is valid for all $ n \geq 2 $, as the result in $ \mathbb{R}^4 $ suffices to disprove the conjecture in higher dimensions via standard extension arguments.
• The minimal $ T^\circ $-length of a 3-vertex billiard trajectory in $ K $ is achieved by a 2-bounce trajectory along a diagonal, and this value equals the Ekeland–Hofer–Zehnder capacity.
• The result implies that symplectic capacities are not uniquely determined by volume on the class of convex domains, and that the Euclidean ball does not maximize symplectic size among all convex bodies of equal volume.
• A smooth approximation of the pentagon-based domain yields a $ C^\infty $ convex body that also violates Viterbo’s conjecture, showing the failure is not limited to polyhedral or non-smooth domains.