  For a given polarized toric variety, we define the notion of
$\lambda$-stability which is a natural generalization of uniform K-stability.
At the neighbourhoods of the vertices of the corresponding moment polytope
$\Delta$, we consider appropriate triangulations and give a sufficient criteria
for a $\lambda$-stable polarized toric variety $(X,L)$ to be asymptotically
Chow polystable when the obstruction of asymptotic Chow semistability (the
Futaki-Ono invariant) vanishes. As an application, we prove that any
K-semistable polarized smooth toric variety $(X,L)$ with the vanishing
Futaki-Ono invariant is asymptotically Chow polystable.
