Abstract
Schuster introduced the notion of radial Blaschke-Minkowski homomorphism and considered the Busemann-Petty problem for volume forms. Whereafter, Wang, Liu and He presented the L p radial Blaschke-Minkowski homomorphisms and extended Schuster’s results. In this paper, associated with L p dual affine surface areas, we give an affirmative and a negative form of the Busemann-Petty problem and establish two Brunn-Minkowski inequalities for the L p radial Blaschke-Minkowski homomorphisms.
Highlights
If K is a compact star shaped in n-dimensional Euclidean space Rn, its radial function, ρK = ρ(K, ·) : Rn \{0} → [0, ∞), is defined byρ(K, x ) = max{λ ≥ 0 : λx ∈ K }, x ∈ Rn \{0}.If ρ(K, ·) is positive and continuous, K will be called a star body
For K ∈ Son, the intersection body, IK, of K is a star body whose radial function is defined by ρ( IK, u) = Vn−1 (K ∩ u⊥ )
In 2015, Wang and Wang [35] defined the L p radial Blaschke combinations of star bodies as follows: For K, L ∈ Son, n > p > 0 and λ, μ ≥ 0, the L p radial Blaschke combination, b p μ ◦ L ∈ Son, of K and L is defined by λ ◦ K+
Summary
For K ∈ Son , the intersection body, IK, of K is a star body whose radial function is defined by ρ( IK, u) = Vn−1 (K ∩ u⊥ ). Intersection bodies led to the following Busemann-Petty problem (see [2]). Blaschke-Minkowski homomorphism, which is the more general intersection operator as follows: Definition 1. Whereafter, Schuster ([22]) considered the following Busemann-Petty problem for radial. We improve Wang, Yuan and He’s definition as follows: For K ∈ Son and p > 0, the L p dual affine e p (K ), of K is defined by surface area, Ω p e p (K ). Associated with Lq radial Minkowski sum and Lq harmonic Blaschke sum of star bodies, we establish the following L p dual affine surface area forms of Brunn-Minkowski inequalities for the.
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