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.
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