Determine the maximal symplectic systolic constant for convex domains; even the Lagrangian product case with one Euclidean ball remains open

Determine the value of Sys(K^{2n}) = sup_{K in K^{2n}} c_EHZ(K) / (n! Vol(K))^{1/n} for convex domains K ⊂ R^{2n}, where c_EHZ denotes the Ekeland–Hofer–Zehnder capacity. In particular, determine this value when restricted to Lagrangian products K × T ⊂ R_q^n × R_p^n, even in the special case where one of the factors is the Euclidean ball.

Background

The paper defines the maximal symplectic systolic constant Sys(K^{2n}) for convex domains in R^{2n} via the Ekeland–Hofer–Zehnder capacity, normalized by volume. Prior work shows that this ratio is bounded above by a universal constant for convex domains, unlike the star-shaped case where it is unbounded. It is also known that Sys(K^{2n}) is nondecreasing with respect to the ambient dimension.

Motivated by their counterexample to Viterbo’s conjecture, the authors raise the problem of determining the exact value of Sys(K^{2n}). They note that this problem is already challenging and remains open for the sub-class of Lagrangian products K × T in R_q^n × R_p^n, even when one factor is the Euclidean ball.