  In this note, we prove that on polarized toric manifolds the relative
$K$-stability with respect to Donaldson's toric degenerations is a necessary
condition for the existence of Calabi's extremal metrics, and also we show that
the modified $K$-energy is proper in the space of $G_0$-invariant K\"ahler
metrics for the case of toric surfaces which admit the extremal metrics.
