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In this paper, we prove that a Gorenstein toric Fano variety $(X, -K_{X})$ isasymptotically Chow semistable then it is Ding polystable with respect to torictest configurations (Theorem 1.3). This extends the known result obtained byothers (Theorem 1.2) to the case where $X$ admits Gorenstein singularity. Wealso show the additivity of the Mabuchi constant for the product toric Fanovarieties in Proposition 1.5 based on the author's recent work (Ono, Sano andYotsutani in arXiv:2305.05924). Applying this formula to certain toric Fanovarieties, we construct infinitely many examples that clarify the differencebetween relative K-stability and relative Ding stability in a systematic way(Proposition 1.4). Finally, we verify relative Chow stability for Gorensteintoric del Pezzo surfaces using the combinatorial criterion developed in(Yotsutani and Zhou in Tohoku Math. J. 71 (2019), 495-524.) and specifying thesymmetry of the associated polytopes as well.

Asymptotic Chow semistability implies Ding polystability for Gorenstein toric Fano varieties