Complex $L_p$-Intersection Bodies

Interpolating between the classic notions of intersection and polar centroid bodies, (real) $L_p$-intersection bodies, for $-1<p<1$, play an important role in the dual $L_p$-Brunn–Minkowski theory. Inspired by the recent construction of complex centroid bodies, a complex version of $L_p$-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 $L_p$-intersection body of an ${\mathbb{S}}^1$-invariant convex body is pseudo-convex, if $-2<p<-1$ and convex, if $p\ge -1$. Moreover, intersection inequalities of Busemann–Petty type in the sense of Adamczak–Paouris–Pivovarov–Simanjuntak are deduced.