Topology of the Grünbaum-Hadwiger-Ramos hyperplane mass partition problem

Hyperplane mass equipartition problem and the shielding functions of Ramos

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We give a proof (based on methods and ideas developed in [16,9,18]) of the result of Ramos [11] which claims that two finite, continuous Borel measures µ 1 and µ 2 defined on R 5 admit an equipartition by a collection of three hyperplanes. Our proof illuminates one of the central methods developed and used in our earlier papers and may serve as a good 'test case' for addressing (and resolving) the 'issues' raised in [2]; see Sections 1 and 4 for an outline and summary. We also offer a degree-theoretic interpretation of the 'parity calculation method' developed in [11] and demonstrate that, up to minor corrections or modifications, it remains a rigorous and powerful tool for proving results about mass equipartitions.

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“…The Grünbaum-Hadwiger-Ramos hyperplane mass equipartition problem [6,7,1,11,16,9,18,19,2,3] has been for decades one of the important test problems for applications of topological methods in discrete geometry.…”

“…The problem came into the focus again with the appearance of the 'critical review' [2] which, as claimed by the authors, included the 'documentation of essential gaps' in the proofs of some of the earlier papers. In turn this led to an interesting and important academic discussion about the validity, scope and applicability of the previously used methods.…”

“…In this paper we address the central objections raised in [2] (the reader will find a brief summary in Section 1.1 and concluding comments in Section 4).…”

Measure partitions via Fourier analysis II: center transversality in the $${{L}^{2}}$$ L 2 -norm for complex hyperplanes

“…. , R m } by a fixed type of "nice" convex regions -e.g., those determined by affine independent or pairwise orthogonal hyperplane collections [3,4,6,11,12,18,22,28], arrangements by more general fans [1,2,17], cones on polytopes [15,26,31], et cetera -so that each region contains an equal fraction of each total measure: µ j (R i ) = µ j (R d )/m for all 1 ≤ i ≤ m and all 1 ≤ j ≤ n. On the other hand are the center-transversality questions, in which the goal is to find an affine space A of a specified dimension such that each partition P with center containing A comes "sufficiently close" to equipartitioning each measure. For instance, the classical center-point theorem of Rado [21] claims for any mass µ on R d the existence of a point p so that |µ(H ± ) − µ(R d )/2| ≤ µ(R d )/6 for the half-spaces H ± of any hyperplane H passing through p, and moreover that the ratio 1/6 is minimal over all masses.…”