  In this note, we study the asymptotic Chow stability of toric varieties. We
provide examples of symmetric reflexive toric varieties that are not asymptotic
Chow semistable. On the other hand, we also show that any weakly symmetric
reflexive toric varieties which have regular triangulation (special) are
asymptotic Chow polystable.
  After that, we provide another criteria that can show a symmetric reflexive
toric variety is asymptotic Chow polystable. In particular, we give two
examples that are asymptotic Chow polystable, but not special. We also provide
some examples of special polytopes, mainly in 2 or 3 dimensions, and some in
higher dimensions.
