  We express notions of K-stability of polarized spherical varieties in terms
of combinatorial data, vastly generalizing the case of toric varieties. We then
provide a combinatorial sufficient condition of G-uniform K-stability by
studying the corresponding convex geometric problem. Thanks to recent work of
Chi Li and a remark by Yuji Odaka, this provides an explicitly checkable
sufficient condition of existence of constant scalar curvature Kahler metrics.
As a side effect, we show that, on several families of spherical varieties,
G-uniform K-stability is equivalent to K-polystability with respect to
G-equivariant test configurations for polarizations close to the anticanonical
bundle.
