Remarks on symplectic capacities of p-products

In this paper, 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 nontrivial p-decompositions of the symplectic ball.