  In this note we apply the billiard technique to deduce some results on
Viterbo's conjectured inequality between volume of a convex body and its
symplectic capacity. We show that the product of a permutohedron and a simplex
(properly related to each other) delivers equality in Viterbo's conjecture.
Using this result as well as previously known equality cases, we prove some
special cases of Viterbo's conjecture and interpret them as isoperimetric-like
inequalities for billiard trajectories.
