  We prove that among all Koll\'ar components obtained by plt blow ups of a klt
singularity $o \in (X, D)$, there is at most one that is (log-)K-semistable. We
achieve this by showing that if such a Koll\'ar component exists, it uniquely
minimizes the normalized volume function introduced in [Li15a] among all
divisorial valuations. Conversely, we show any divisorial minimizer of the
normalized volume function yields a K-semistable Koll\'ar component. We also
prove that for any klt singularity, the infimum of the normalized function is
always approximated by the normalized volumes of Koll\'ar components.
