  In this work we investigate subsets of $\mathbb{R}^{2n}$ with the property
that their mean width can not be decreased by the action of natural classes of
symplectomorphisms. A common theme of our results is that toric symmetry is a
preferred feature of subsets that are in optimal symplectic position with
respect to the mean width. Among other things, we prove that the mean width of
a toric domain can not be decreased by any linear symplectic map. We also show
that toric convex domains are critical points of the mean width under all
symplectic deformations.
