2-Dimension Ham Sandwich Theorem for Partitioning into Three Convex Pieces

Ito, Hiro; Uehara, Hiroki; Yokoyama, Mitsuru

doi:10.1007/978-3-540-46515-7_11

Cited by 36 publications

(25 citation statements)

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“…Similar results, which seems to be essentially equivalent to the following 3-cutting Theorem, were obtained in [39] and [19], respectively. …”

Section: Lemma 31 (The Intermediate Line Lemma) Let K ≥ 0 Be An Intsupporting

confidence: 83%

“…The following theorem, which was conjectured in [21] and proved for a = 1, 2 in [21] and [26], was completely proved by Bespamyatnikh, Kirkpatrick and Snoeyink [7], Sakai [39] and by Ito, Uehara and Yokoyama [19] independently. Note that for g = 2, it is equivalent to the Ham-Sandwich Theorem above.…”

Section: Draftmentioning

confidence: 95%

“…Note that for g = 2, it is equivalent to the Ham-Sandwich Theorem above. [7], [39], [19]). Let a ≥ 1, b ≥ 1 and g ≥ 2 be integers.…”

Section: Draftmentioning

confidence: 99%

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Discrete Geometry on Red and Blue Points in the Plane — A Survey —

Kaneko

Kanō

2003

Algorithms and Combinatorics

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In this paper, we give a short survey on discrete geometry on red and blue points in the plane, most of whose results were obtained in the past decade. We consider balanced subdivision problems, geometric graph problems, graph embedding problems, Gallai-type problems and others. Notation and DefinitionsIn this paper, we give a short survey on discrete geometry on red and blue points in the plane, most of whose results were obtained in the past decade. We consider two disjoint sets R and B of red points and of blue points in the plane, respectively, such that no three points of R ∪ B lie on the same line. Throughout this paper, R and B always denote the sets mentioned above unless noted otherwise.We begin with some notation and definitions, which will be used throughout this paper. A (directed) line l trisects the plane into three pieces: l, right(l) and lef t(l), where right(l) and lef t(l) denote the open half-planes which are on the right side and on the left side of l, respectively (Figure 1). Let r 1 and r 2 be two rays emanating from the same point p. Then we denote by right(r 1 ) ∩ lef t(r 2 ) the open region that is swept by the ray being rotated clockwise around p from r 1 to r 2 (Figure 1). The open region lef t(r 1 ) ∩ right(r 2 ) is similarly defined. Then r 1 ∪ r 2 trisects the plane into three pieces: r 1 ∪ r 2 and two open regions right(r 1 ) ∩ lef t(r 2 ) and lef t(r 1 ) ∩ right(r 2 ). If the internal angle ∠r 1 pr 2 = ∠r 1 r 2 of right(r 1 ) ∩ lef t(r 2 ) is less than π, then we call right(r 1 ) ∩ lef t(r 2 ) the wedge defined by r 1 and r 2 , and denote it by wedge(r 1 r 2 ), wedge(r 2 r 1 ), wedge(r 1 pr 2 ) or wedge(r 2 pr 1 ). left(l)right ( Given a point p and a line l passing through p, let l * denote the line lying on l and having the opposite direction of l. We define the rays r and r * to be the two rays emanating from p and lying on the line l, while having the direction of l and l * respectively (Figure 1). Conversely, given a ray r, we can define r * , l and l * .For a set X of points in the plane, let conv(X) denote the convex hull of X, which is the smallest convex set containing X. For convenience, a region in the plane whose boundary consists of straightline segments is called a polygon even if it is an infinite region. For example, Figure 2

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“…Similar results, which seems to be essentially equivalent to the following 3-cutting Theorem, were obtained in [39] and [19], respectively. …”

Section: Lemma 31 (The Intermediate Line Lemma) Let K ≥ 0 Be An Intsupporting

confidence: 83%

Section: Draftmentioning

confidence: 95%

See 1 more Smart Citation

Discrete Geometry on Red and Blue Points in the Plane — A Survey —

Kaneko

Kanō

2003

Algorithms and Combinatorics

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“…Very recently the conjecture of Kaneko and Kano has been independently proven by Ito et al [10] and Sakai [16]. Comparing our result with papers [10] and [16] we can say that they prove essentially the same 3-cutting theorems but both use different (and somewhat more complicated) techniques.…”

Section: Theorem 2 (3-cutting) For Any Gn Red Points (G~2) and Gm Blsupporting

confidence: 68%

Generalizing Ham Sandwich Cuts to Equitable Subdivisions

Bespamyatnikh

Kirkpatrick

Snoeyink

2000

Discrete Comput Geom

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Abstract. We prove a generalization of the famous Ham Sandwich Theorem for the plane.Given gn red points and gm blue points in the plane in general position, there exists an equitable subdivision of the plane into g disjoint convex polygons, each of which contains n red points and m blue points. For g = 2 this problem is equivalent to the Ham Sandwich Theorem in the plane. We also present an efficient algorithm for constructing an equitable subdivision.

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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%

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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2-Dimension Ham Sandwich Theorem for Partitioning into Three Convex Pieces

Cited by 36 publications

References 1 publication

Discrete Geometry on Red and Blue Points in the Plane — A Survey —

Discrete Geometry on Red and Blue Points in the Plane — A Survey —

Generalizing Ham Sandwich Cuts to Equitable Subdivisions

Computational topology of equivariant maps from spheres to complements of arrangements

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