Two Complex Combinations and Complex Intersection Bodies

Abstract

This paper devotes to establish complex dual Brunn-Minkowski theory. At first, we introduce the concepts of complex radial combination and complex radial-Blaschke combination, and obtain the relations between those two combinations and dual mixed volumes. Then, we extend the properties of real intersection body to the complex case. Finally, we prove some complex geometric inequalities about complex intersection bodies and complex mixed intersection bodies, such as dual Brunn-Minkowski type, dual Aleksandrov-Fenchel type and dual Minkowski type inequality. Moreover, as applications, we get some corollaries including an isoperimetric type inequality and a uniqueness theorem.

Dual mixed complex brightness integrals

“…From these, and according to (35), (36) and 34 By the equality conditions of inequalities (35) and 34, we see that equality holds in (33)…”

“…Until recently, the situation with complex convex bodies began to attract attention (see [2, 4, 12-15, 17, 26, 42, 43]). Some classical concepts of convex geometry in real vector space were extended to complex cases, such as complex projection bodies (see [3,20,29,39]), complex difference bodies (see [1]), complex intersection bodies (see [16,30,36,40]), complex centroid bodies (see [10,19]) and mixed complex brightness integrals (see [18]).…”

“…In particular, by I C B = ω 2n−2 π B (see page 1642 of [36]), since for every ξ ∈ S 2n−1 , by (2), Wang et.al. (see page 422 of [30]) obtained…”

“…Above definition (20) was extended to the complex case by Wu et al (see [36]). For K, L ∈ S(C n ) and λ, µ ≥ 0 (not both zero), the complex radial Blaschke linear combination, λ • K + C µ • L, of K and L as the star body whose radial function is given by:…”