Measure Partitions via Fourier Analysis II: Center Transversality in the $L^2$-norm for Complex Hyperplanes

TL;DR

This paper extends harmonic analysis methods to the setting of compact Lie groups to establish measure transversality for complex hyperplanes, generalizing classical equipartition results using Fourier analysis.

Contribution

It introduces a novel approach applying continuous group actions and Fourier series to measure partition problems, expanding beyond finite groups.

Findings

Existence of complex hyperplanes with measure partitions close in L^2-norm

First use of continuous group actions in equivariant topological methods for geometry

Generalization of classical centerpoint theorem to complex hyperplanes

Abstract

Applications of harmonic analysis on finite groups were recently introduced to measure partition problems, with a variety of equipartition types by convex fundamental domains obtained as the vanishing of prescribed Fourier transforms. Considering the circle group, we extend this approach to the compact Lie group setting, in which case the annihilation of transforms in the classical Fourier series produces measure transversality similar in spirit to the classical centerpoint theorem of Rado: for any $q\geq 2$, the existence of a complex hyperplane whose surrounding regular $q$-fans are close -- in an $L^2$-sense -- to equipartitioning a given set of measures. The proofs of these results represent the first application of continuous as opposed to finite group actions in the usual equivariant topological reductions prevalent in combinatorial geometry.