Complex Lp-intersection bodies

Abstract

Interpolating between the classical notions of intersection and polar centroid bodies, (real) Lp-intersection bodies, for −1<p<1, play an important role in the dual Lp-Brunn–Minkowski theory. Inspired by the recent construction of complex centroid bodies, a complex version of Lp-intersection bodies, with range extended to p>−2, is introduced, interpolating between complex intersection and polar complex centroid bodies. It is shown that the complex Lp-intersection body of an S1-invariant convex body is pseudo-convex, if −2<p<−1 and convex, if p≥−1. Moreover, intersection inequalities of Busemann–Petty type in the sense of Adamczak–Paouris–Pivovarov–Simanjuntak are deduced.