Abstract
The notion of intersection body is introduced by Lutwak in 1988, it is one of important research contents and led to the studies of Busemann-Petty problem in the Brunn-Minkowski theory. Based on the properties of the intersection bodies, Schuster introduced the notion of radial Blaschke-Minkowski homomorphisms and proved a lot of related inequalities. In this paper, by applying the dual mixed volume theory and analytic inequalities, we first give a lower bound of the dual quermassintegrals for the mixed radial Blaschke-Minkowski homomorphisms. As its an application, we get a reverse form of the well-known Busemann intersection inequality. Further, a Brunn-Minkowski type inequality of the Lp radial Minkowski sum for the dual quermassintegrals of mixed radial Blaschke-Minkowski homomorphisms is established, and then the intersection body version of this Brunn-Minkowski type inequality is yielded. From this, we not only extend Schuster's related result but also obtain the Brunn-Minkowski type inequalities of Lp harmonic radial sum and Lp radial Blaschke sum, respectively.
Highlights
The setting for this paper is Euclidean n-space Rn
Let Son denote the set of all star bodies in Rn
Blaschke-Minkowski homomorphism, together with the radius of IB is ωn−1, Corollary 1.1 provides the following a new inequality for the volume of intersection body
Summary
The setting for this paper is Euclidean n-space Rn. Let Sn−1 denote the unit sphere in Rn and V (K) denote the ndimensional volume of body K. Schuster [10] proved the following important result: Weidong Wang: Inequalities for the Mixed Radial Blaschke-Minkowski Homomorphisms and the Applications. Blaschke-Minkowski homomorphism, together with the radius of IB is ωn−1, Corollary 1.1 provides the following a new inequality for the volume of intersection body. Let i = j = 0 in Theorem 1.2, and notice intersection operator I is a special case of radial Blaschke-Minkowski homomorphisms, we have Corollary 1.3 Let K, L ∈ Son, real p = 0. Brunn-Minkowski type inequalities for the Lq harmonic radial sum and the Lq radial Blaschke sum as follows: Corollary 1.4 If K, L ∈ Son, real q ≥ 1, i = 0, 1, · · · , n − 2 and j = 0, 1, · · · , n − 2, . Corollary 1.4 and Corollary 1.5 were established by Wei, Wang and Lu [19]
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