Coincidence of ball-normalized capacities on systolically convex domains in T*2

Prove that for every fiberwise star-shaped domain X ⊂ (T*2, dλ_can) that is systolically convex—meaning X is dynamically convex and its systolic ratio ρ_sys,T*2(X) := sys(∂X)^2 / vol(∂X, λ_can|∂X) satisfies ρ_sys,T*2(X) ≤ 1/4—all ball-normalized symplectic capacities assign the same value to X (i.e., every pair of ball-normalized capacities coincide on X).

Background

The paper introduces the subclass S of fiberwise star-shaped domains in (T*2, dλ_can) called systolically convex, defined by dynamical convexity together with an upper bound on the systolic ratio ρ_sys,T*2 ≤ 1/4. This class is proposed as a cotangent-bundle analogue of geometrically convex domains in Euclidean symplectic spaces.

Prior work has established coincidence of ball-normalized capacities for several specific families within T*2 and T*n (e.g., codisc bundles of flat reversible Finsler metrics, certain product domains corresponding to monotone toric profiles), while in Euclidean space the strong Viterbo conjecture has recently been disproved for convex domains. Motivated by these mixed outcomes, the authors formulate a conjecture that aims to characterize a robust regime—namely S—where normalized capacities might universally agree in the T*2 setting.

References

Conjecture. All ball-normalized capacities coincide for elements in \mathcal S.

— Geometry and dynamics on Liouville domains in $T^*\mathbb T^2$  (2603.29253 - Zhang et al., 31 Mar 2026) in Introduction, Conjecture (Conjecture T-Vit)