Lp-mixed affine surface area

According to the three notions of mixed affine surface area, Lp-affine surface area and Lp-mixed affine surface area proposed by Lutwak, in this article, we give the concept of ith Lp-mixed affine surface area such that the first and second notions of Lutwak are its special cases. Further, some Lutwak's results are extended associated with this concept. Besides, applying this concept, we establish an inequality for the volumes and dual quermassintegrals of a class of star bodies.