For a polarized Kähler manifold
(
X
,
L
)
(X,L)
, we show the equivalence between relative balanced embeddings introduced by Mabuchi and
σ
\sigma
-balanced embeddings introduced by Sano, answering a question of Hashimoto. We give a GIT characterization of the existence of a
σ
\sigma
-balanced embedding, and relate the optimal weight
σ
\sigma
to the action of
A
u
t
0
(
X
,
L
)
\mathrm {Aut}_0(X,L)
on the Chow line of
(
X
,
L
)
(X,L)
.