Motivated by some results from differential geometry, I will discuss a purely algebraic problem of minimizing the normalizedvolume functional on valuation spaces at Kawamata log-terminal (Klt) singularities. In the case when the minimizers are associated to K-semistable log Fano cones (conjectured to be always true), we can prove that theyare unique among all quasi-monomial valuations. Moreover, any K-semistable log Fano cone degenerates uniquely to a K-polystable log Fano cone. This allows us to attach canonical semistable/polystable objects and volume-type invariant to any Klt singularity. As one application, we answer Donaldson-Sun's conjecture about algebraicity of metric tangent cones on Gromov-Hausdorff limits. This talk is mostly based on joint works with Chenyang Xu.