$q$-Quermassintegrals form of the polar duality of $L_p$-Brunn–Minkowski inequality and the applications

The quermassintegrals form of the polar duality of Brunn–Minkowski inequality was established by Firey. In 2016, Cifre and Nicolas gave the quermassintegrals form of the polar duality of $L_p$-Brunn–Minkowski inequality. In this paper, we extend Cifre and Nicolas's result to $q$-quermassintegrals form. Further, as the applications of this result, we separately give the $q$-quermassintegrals form of the $L_p$-dual Brunn–Minkowski inequality and the extremums of $q$-quermassintegrals of the polar asymmetric $L_p$-difference bodies.