Optimal destabilizing centers and equivariant K-stability

We give an algebraic proof of the equivalence of equivariant K-semistability (resp. equivariant K-polystability) with geometric K-semistability (resp. geometric K-polystability). Along the way we also prove the existence and uniqueness of minimal optimal destabilizing centers on K-unstable log Fano pairs.Comment: 23 pages. v3: minor revision following referee's comments. To appear in Invent. Mat