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

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