The Gromov width of a symplectic manifold measures the largest ball
that can be symplectically embedded in it. We find a lower bound for
the Gromov width of regular coadjoint orbits of the symplectic group,
which coincides with the previously known upper bound. This is
achieved via a toric degeneration, in which the resulting Okounkov
body is a rational polytope. Kaveh has shown that it coincides with
the string polytope corresponding to the dual crystal basis for a
certain representation of the symplectic group, lending itself to
concrete computations towards the result. This is joint work with
Milena Pabiniak.