  In this paper, we discuss the relative $K$-stability and the modified
$K$-energy associated to the Calabi's extremal metric on toric manifolds. We
give a sufficient condition in the sense of convex polytopes associated to
toric manifolds for both the relative $K$-stability and the properness of
modified $K$-energy. In particular, our results hold for toric Fano manifolds
with vanishing Futaki-invariant. We also verify our results on the toric Fano
surfaces.
