Relative Chow stability and optimal weights

For a polarized K"ahler manifold $(X,L)$, we show the equivalence between relative balanced embeddings introduced by Mabuchi and $\sigma$-balanced embeddings introduced by Sano, answering a question of Hashimoto. We give a GIT characterization of the existence of a $\sigma$-balanced embedding, and relate the optimal weight $\sigma$ to the action of ${\mathrm{A}\mathrm{u}\mathrm{t}}_0(X,L)$ on the Chow line of $(X,L)$.