The complex Busemann-Petty problem for arbitrary measures

TL;DR

This paper extends the complex Busemann-Petty problem to arbitrary measures, showing the same dimensional threshold results as the volume case, thus generalizing previous work to a broader measure context.

Contribution

It proves that the known dimension-dependent affirmative and negative answers hold when volume is replaced by an almost arbitrary measure in the complex setting.

Findings

Affirmative for n ≤ 3

Negative for n ≥ 4

Generalizes to arbitrary measures

Abstract

The complex Busemann-Petty problem asks whether origin symmetric convex bodies in C^n with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if n\leq 3 and negative if n\geq 4. In this article we show that the answer remains the same if the volume is replaced by an "almost" arbitrary measure. This result is the complex analogue of Zvavitch's generalization to arbitrary measures of the original real Busemann-Petty problem.