Complex Intersection Bodies

TL;DR

This paper introduces complex intersection bodies, extending properties of real intersection bodies to the complex setting, including convexity and stability results, with implications for geometric inequalities and the Busemann-Petty problem.

Contribution

It generalizes Busemann's theorem to complex intersection bodies and establishes new stability and hyperplane inequalities in the complex convex geometry context.

Findings

Complex intersection bodies are convex for symmetric complex convex bodies.

Stability results are proved for the complex Busemann-Petty problem.

Hyperplane inequalities are established for measures of complex intersection bodies.

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.