Inequalities for dual quermassintegrals of mixed intersection bodies

In this paper, we first introduce a new concept of dual quermassintegral sum function of two star bodies and establish Minkowski's type inequality for dual quermassintegral sum of mixed intersection bodies, which is a general form of the Minkowski inequality for mixed intersection bodies. Then, we give the Aleksandrov– Fenchel inequality and the Brunn–Minkowski inequality for mixed intersection bodies and some related results. Our results present, for intersection bodies, all dual inequalities for Lutwak's mixed prosection bodies inequalities. One might say the history of intersection bodies began with the paper of Busemann [4]. Intersection bodies were first explicitly defined and named by Lutwak [11]. It was here that the duality between intersection bodies and projection bodies was first made clear. Despite considerable ingenuity of earlier attacks on the Busemann–Petty problem, it seems fair to say that the work of Lutwak [11] represents the beginning of its eventual solution. In [11], Lutwak also showed that if a convex body is sufficiently smooth and not an intersection body, then there exists a centred star body such that the conditions of Busemann–Petty problem holds, but the result inequality is reversed. Following Lutwak, the intersection body of order i of a star body is introduced by Zhang [21]. It follows from this definition that every intersection body of order i of a star body is an intersection body of a star body, and vice versa. As Zhang observes, the new definition of intersection body allows a more appealing formulation, namely: the Busemann–Petty problem has a positive answer in n-dimensional Euclidean space if and only if each centered convex body is an intersection body. The intersection body plays an essential role in Busemann's theory [5] of area in Minkowski spaces. The intersection body is also an important matter of the Brunn–Minkowski theory. have given considerable attention to the Brunn–Minkowski theory and their various generalizations. The purpose of this paper is to establish the Minkowski inequality for the dual quermassintegral sum, which is a generalization of the Minkowski inequality for mixed intersection bodies. Then, 79