  Let G be a compact connected Lie group G and T its maximal torus. The
coadjoint orbit O_lambda through lambda in Lie(T)^* is canonically a symplectic
manifold. Therefore we can ask the question about its Gromov width. In many
known cases the Gromov width is exactly the minimum over the set {<
alpha_j^{\vee},lambda > ; alpha_j^{\vee} a coroot and < alpha_j^{\vee},lambda >
positive}. We show that the Gromov width of coadjoint orbits of the unitary
group and of most of the coadjoint orbits of the special orthogonal group is at
least the above minimum. The proof uses the torus action coming from the
Gelfand-Tsetlin system.
