Remarks on symplectic capacities of $p$-products

TL;DR

This paper investigates how symplectic capacities behave under symplectic p-products of convex domains, and connects these properties to Viterbo's volume-capacity conjecture and higher-order capacities.

Contribution

It introduces a new approach using the tensor power trick to relate the conjecture to asymptotic regimes and proposes a conjecture on higher-order capacities impacting p-decompositions.

Findings

Weak version of Viterbo's conjecture holds asymptotically

No non-trivial p-decompositions of the symplectic ball if the conjecture on higher-order capacities is true

Symplectic capacities exhibit specific behaviors under p-products

Abstract

In this note we study the behavior of symplectic capacities of convex domains in the classical phase space with respect to symplectic $p$-products. As an application, by using a "tensor power trick", we show that it is enough to prove the weak version of Viterbo's volume-capacity conjecture in the asymptotic regime, i.e., when the dimension is sent to infinity. In addition, we introduce a conjecture about higher-order capacities of $p$-products, and show that if it holds, then there are no non-trivial $p$-decompositions of the symplectic ball.