Equipartition of mass distributions by hyperplanes

Transversals of d-Intervals

“…These We easily obtain an analogue of Lemma 2.2. established by Ramos, and refer the reader to [7] for a proof. The original version deals with products of balls rather than spheres, but it is equivalent to the one we will use.…”

Equipartitions of measures in $\mathbb{R}^4$

“…D. Avis showed in [Avis84] that there exist non-equipartitionable mass distributions for n ≥ 5. In the meantime many related questions were formulated and many of them solved ( [BM01], [BM02], [Mak01], [Ram96], [VZ92]), a connection with discrete and computational geometry was established ( [YY85], [YDEP89]) and the subject grew 154 RADE T.ŽIVALJEVIĆ into a separate branch of geometric combinatorics [Živ04]. Nevertheless, the 4-equipartition problem itself has resisted all attempts and remains one of the central open problems in the field.…”

“…A well-known problem of B. Grünbaum (1960) asks whether for every continuous mass distribution (measure) dµ = f dm on R n there exist n hyperplanes dividing R n into 2 n parts of equal measure. It is known that the answer is positive in dimension n = 3 (see H. Hadwiger (1966)) and negative for n ≥ 5 (see D. Avis (1984) andE. Ramos (1996)).…”

Conical equipartitions of mass distributions

“…Surveyarticles [8], [14], [23]- (25] as well as the original papers [18], [21], and [27] coverdifferent aspectsof the equipartitionproblemand give a good picture of someof the more recentdevelopmentsin this area. For example, Ramosin (18] coversthe questionof equipartitionsof masses by hyperplanes, Vrecica andZivaljevic [21] dealwith equipartitionsby regularwedge-likecones,andthe "center transversal theorem"provedin [27] revealsa hiddenconnectionbetweenthe center point theoremand the ham sandwichtheorem,etc. On the other hand, some of the key "divide and conquer"paradigmsfor designingalgorithms in Computational Geometryrely on the existenceof "balanced"partitions for different classesof geometric objects.For example,Griinbaumraisedin [12] a questionwhether,for a given measuref.L in R d , therealwaysexist d hyperplanessuchthat eachof the corresponding open orthantscontainsat most a fraction 1/2 d of the massf.L(R d ).…”

“…Surveyarticles [8], [14], [23]- (25] as well as the original papers [18], [21], and [27] coverdifferent aspectsof the equipartitionproblemand give a good picture of someof the more recentdevelopmentsin this area. For example, Ramosin (18] coversthe questionof equipartitionsof masses by hyperplanes, Vrecica andZivaljevic [21] dealwith equipartitionsby regularwedge-likecones,andthe "center transversal theorem"provedin [27] revealsa hiddenconnectionbetweenthe center point theoremand the ham sandwichtheorem,etc.…”