Inequalities for Mixed Complex Projection Bodies

Abstract

Complex projection bodies were introduced by Abardia and Bernig, recently. In this paper some geometric inequalities for mixed complex projection bodies which are analogs of inequalities for mixed real projection bodies are established.

Dual mixed complex brightness integrals

“…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]).…”

“…for i > 2n − 1 and i 2n, inequality (29) is reversed. Equality holds in (29) if and only if M and N have similar dual complex brightness and (…”

“…By the equality conditions of inequalities (11) and (30), we see that equality holds in (29) if and only if M and N have similar dual complex brightness and there exists constant λ such that (…”