Inequalities for General Width-Integrals of Blaschke-Minkowski Homomorphisms
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Czechoslovak Mathematical Journal
13 pp
Online first
INEQUALITIES FOR GENERAL WIDTH-INTEGRALS OF BLASCHKE-MINKOWSKI HOMOMORPHISMS Chao Li, Weidong Wang, Yichang Received November 27, 2018. Published online January 27, 2020.
Abstract. We establish some inequalities for general width-integrals of BlaschkeMinkowski homomorphisms. As applications, inequalities for width-integrals of projection bodies are derived. Keywords: general width-integral; volume difference type inequality; Blaschke-Minkowski homomorphism; Brunn-Minkowski type inequality; projection body MSC 2010 : 52A20, 52A40
1. Introduction Let Kn denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space Rn . The n-dimensional volume of the body M ∈ Kn is denoted by V (M ). For the standard unit ball U , write V (U ) = ωn . The unit sphere, i.e. the boundary of U , is denoted by S n−1 and the surface area measure on S n−1 is denoted by S(·). A convex body M ∈ Kn is uniquely determined by its support function h(M, ·) : Rn → R, which is defined by h(M, x) = max{x · y : y ∈ M },
x ∈ Rn ,
where x · y denotes the standard inner product of x and y in Rn . For M, N ∈ Kn and λ, µ > 0 (not both zero), the Minkowski linear combination λM + µN of M and N is defined by λM + µN = {λx + µy : x ∈ M, y ∈ N }, Research is supported in part by the Natural Science Foundation of China (No. 11371224) and the Innovation Foundation of Graduate Student of China Three Gorges University (No. 2019SSPY146). DOI: 10.21136/CMJ.2020.0521-18
1
which is equivalent to (1.1)
h(λM + µN, ·) = λh(M, ·) + µh(N, ·).
We refer to the extensive monographs (see [12], [24]) for more background on convex geometry. Width-integrals were first proposed by Blaschke, see [3]. In 1975, Lutwak in [20] introduced width-integrals of index i and the mixed width-integral for convex bodies, see [21]. In 2010, Lv in [23] studied the width-integral difference. Later on, Zhao and Mihály in [36] established some Brunn-Minkowski inequalities for width-integrals of mixed projection bodies. In 2016, Feng in [6] introduced the concept of general mixed width-integrals for convex bodies, and established the inequality of isoperimetric type, the AleksandrovFenchel type inequality and the cyclic inequality. He also considered the general width-integrals of order i and showed its related properties and inequalities. Recently, Zhou in [37] researched the general Lp -mixed width-integrals of convex bodies, and gave its extremal values and extended Feng’s results. See [4], [9], [18], [30] for more related results on the width-integrals of convex bodies. For M1 , . . . , Mn ∈ Kn , τ ∈ (−1, 1), the general mixed width-integral B (τ ) (M1 , . . . , Mn ) of M1 , . . . , Mn is defined by (1.2)
B
(τ )
1 (M1 , . . . , Mn ) = n
Z
b(τ ) (M1 , u) . . . b(τ ) (Mn , u) dS(u), S n−1
where (1.3)
b(τ ) (M, u) = f1 (τ )h(M, u) + f2 (τ )h(M, −u)
for all u ∈ S n−1 and the functions f1 (τ ) and f2 (τ ) are given by (1.4)
f1 (τ ) =
(1 + τ )2 , 2(1 + τ 2 )
f2 (τ ) =
(1 − τ )2 . 2(1 + τ
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