Radial Perfect Partitions of Convex Sets in the Plane

Akiyama, Jin; Kaneko, Atsuhi; Kanō, Mikio; Nakamura, Gisaku; Rivera-Campo, Eduardo; Tokunaga, Shinichi; Urrutia, Jorge

doi:10.1007/978-3-540-46515-7_1

Cited by 15 publications

(26 citation statements)

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“…Recall that this question arose in connection with some partition problems in discrete and computational geometry, [1], [5], [11], [12], [19] [22]. It turns out that Problem 1 is closely related to a problem of the existence of equivariant maps, see Problem 3 in Section 2.2.…”

Section: The Motivating Problemmentioning

confidence: 99%

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Computational topology of equivariant maps from spheres to complements of arrangements

Blagojević

Vrećica²,

Živaljević

2008

Trans. Amer. Math. Soc.

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The problem of the existence of an equivariant map is a classical topological problem ubiquitous in topology and its applications. Many problems in discrete geometry and combinatorics have been reduced to such a question and many of them resolved by the use of equivariant obstruction theory. A variety of concrete techniques for evaluating equivariant obstruction classes are introduced, discussed and illustrated by explicit calculations. The emphasis is on D 2 n D_{2n} -equivariant maps from spheres to complements of arrangements, motivated by the problem of finding a 4 4 -fan partition of 2 2 -spherical measures, where D 2 n D_{2n} is the dihedral group. One of the technical highlights is the determination of the D 2 n D_{2n} -module structure of the homology of the complement of the appropriate subspace arrangement, based on the geometric interpretation for the generators of the homology groups of arrangements.

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Section: The Motivating Problemmentioning

confidence: 99%

“…Here the arrangement A(α) is the minimal Z n -invariant subspace arrangement generated by L(α) defined by (1).…”

Section: The Obstruction Cocycle C Zn (H)mentioning

confidence: 99%

Computational topology of equivariant maps from spheres to complements of arrangements

Blagojević

Vrećica²,

Živaljević

2008

Trans. Amer. Math. Soc.

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“…1 for examples. Equipartitions find applications in computer science [19] (for example, geometric range searching), in statistics [5] (for example, regression depth) and in "practice" [1] (for example, cutting a cake). [15] conjectured that, for any qn red and qm blue points (q, n, m > 0 are integers) in the plane in general position, there are q disjoint convex polygons with n red and m blue points in each of them.…”

Section: Introductionmentioning

confidence: 99%

Equipartitions of Measures by 2-Fans

Bereg

2005

Discrete Comput Geom

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We study the problem of computing an equitable 2-fan for three masses distributed on the 2-sphere. The existence of an equitable 2-fan was shown by Bárány and Matoušek [3]. The contribution of this paper is twofold. (i) We prove the existence of an infinite set of equitable 2-fans. (ii) We present an efficient algorithm for finding an equitable 2-fan when the mass distributions are discrete, i.e. finite sets of points. Both (i) and (ii) can be easily extended to mass distributions in the plane instead of the sphere.

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“…It was proved in [2] that a perfect partition always exists for every n ≥ 3, that is, the following theorem was obtained. Theorem 1 [2].…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1 [2]. Every convex set in the plane has a perfect n-partition for every integer n ≥ 3 (Fig.…”

Section: Introductionmentioning

confidence: 99%

Perfect Partitions of Convex Sets in the Plane

Kaneko

Kano

2002

Discrete Comput Geom

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For a region X in the plane, we denote by area(X ) the area of X and by (∂(X )) the length of the boundary of X . Let S be a convex set in the plane, let n ≥ 2 be an integer, and let α 1 , α 2 , . . . , α n be positive real numbers such that α 1 + α 2 + · · · + α n = 1 and 0 < α i ≤ 1 2 for all 1 ≤ i ≤ n. Then we shall show that S can be partitioned into n disjoint convex subsets T 1 , T 2 , . . . , T n so that each T i satisfies the following three conditions: (i) area(T i ) = α i × area(S); (ii) (T i ∩ ∂(S)) = α i × (∂(S)); and (iii) T i ∩ ∂(S) consists of exactly one continuous curve.

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Radial Perfect Partitions of Convex Sets in the Plane

Cited by 15 publications

References 1 publication

Computational topology of equivariant maps from spheres to complements of arrangements

Computational topology of equivariant maps from spheres to complements of arrangements

Equipartitions of Measures by 2-Fans

Perfect Partitions of Convex Sets in the Plane

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