A Counterexample to Viterbo's Conjecture

TL;DR

This work provides a counterexample to Viterbo's volume-capacity conjecture for convex domains, showing that symplectic capacities do not necessarily coincide on convex bodies. The authors connect the Ekeland–Hofer–Zehnder capacity of a Lagrangian product $K \times T$ to the minimal $T^{\circ}$-length of closed $T$-billiard trajectories in $K$, and compute an explicit value for a 4-dimensional pentagonal construction: $c_{_{\rm EHZ}}(K \times T) = 2 \cos\left(\frac{\pi}{10}\right)\left(1 + \cos\left(\frac{\pi}{5}\right)\right)$. Extending to higher dimensions via the symplectic 2-product, they show that Viterbo's conjecture fails for all $n \ge 2$. The work also highlights rich connections between Minkowski billiard dynamics, Reeb flows, and symplectic invariants, and raises questions about maximal systolic ratios and the landscape of capacities on convex domains.

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.

Figures (6)

• Figure 1: The pentagon $K$ times the rotated pentagon $T$.
• Figure 2: An example of a $T$-billiard trajectory in $K$.
• Figure 3: The billiard trajectory $\gamma$ in the clockwise orientation case.
• Figure 4: Different minimal $T^\circ$-length billiard trajectories with three bounces (left) and two bounces (right).
• Figure 5: A $T$-billiard trajectory in $K$ of the first kind (right) and its associated unfolding (left).

Theorems & Definitions (12)

• Definition 1.1
• Conjecture 1.2: Viterbo, 2000
• Theorem 1.3
• Proposition 1.4
• proof : Proof of Theorem \ref{['counterexample_thm']}
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Remark 1.8
• Definition 2.1: Minkowski Billiards