Complex intersection bodies

Abstract

Abstract. We introduce complex intersection bodies and show that their properties and applications are similar to those of their real counterparts. In particular, we generalize Busemann's theorem to the complex case by proving that complex intersection bodies of symmetric complex convex bodies are also convex. Other results include stability in the complex Busemann-Petty problem for arbitrary measures and the corresponding hyperplane inequality for measures of complex intersection bodies.

On mixed complex intersection bodies

“…We simply write I C K rather than I C (K) 0 and is called complex intersection body of K, which was defined by Koldobsky et al [20]. Moreover, I C (K, M 1 , .…”

“…Recently, Koldobsky et al [20] first introduced the mixed complex intersection bodies of complex star bodies. Let K 1 , .…”

“…The sum + C denotes the complex radial sum defined by Koldobsky et al [20]: The complex radial sum is a complex star body that has radial function…”

Volume inequalities for complex isotropic measures

“…The idea to find analogs of known results from Euclidean geometry in complex vector spaces is not new. In recent years, the study of convex bodies in C n has received considerable attention (see, e.g., [1][2][3][4][5]13,15,19,[22][23][24][25][26]33,34,37,41,42]). …”

On Polars of Mixed Complex Projection Bodies

“…Until recently the situation with complex convex bodies began to attract attention (see [1,2,9,11,12,13,24,34,36,37]). …”