On the symplectic capacity and mean width of convex bodies

In this note we consider two topics involving the relationship between the symplectic capacity and the mean width of convex bodies in $\mathbb{R}^{2n}$. We first describe an alternative path from the symplectic Brunn-Minkowski inequality of Artstein-Avidan and Ostrover to another inequality, established by the same authors, that relates the capacity and mean width of convex bodies. This new path is less direct but it relates these inequalities to the quermassintegrals of convex bodies and to the local version of Viterbo's conjecture established by Abbondandolo and Benedetti for domains sufficiently close to the ball. We then consider the problem of identifying convex bodies whose mean width cannot be decreased by natural classes of symplectomorphisms. We state a conjectured characterization of convex bodies whose mean width is already minimal among all their symplectic images. To test this conjecture we identify a simple class of quadratic convex bodies whose mean width can not be decreased by linear symplectic maps near the identity. We then identify a subset of these examples that fail to satisfy the toric conditions of the conjecture, and show that one can find a nonlinear symplectomorphism that decreases their mean width.