Uniform K-stability of polarized spherical varieties

We express notions of K-stability of polarized spherical varieties in termsof combinatorial data, vastly generalizing the case of toric varieties. We thenprovide a combinatorial sufficient condition of G-uniform K-stability bystudying the corresponding convex geometric problem. Thanks to recent work ofChi Li and a remark by Yuji Odaka, this provides an explicitly checkablesufficient 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 toG-equivariant test configurations for polarizations close to the anticanonicalbundle.Comment: with an appendix by Yuji Odaka, v2 taking into account referee's comments and correcting minor mistakes, v3 EPIGA journal versio