On the existence of a point subset with a specified number of interior points

Avis, David; Hosono, Kiyoshi; Urabe, Masatsugu

doi:10.1016/s0012-365x(01)00108-x

Cited by 17 publications

(27 citation statements)

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“…In [6], 3 is the smallest positive integer such that any finite point set of at least 3 interior points has a subset for which the interior of the convex hull of the set contains exactly 3 or 4 points in the set .…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…This suffices to show the existence of a deficient point set of type (3,7,3,6). We construct a deficient point set of type (3,7,3,6) as shown in Figure 1. Hence, ℎ(3) ≥ 8.…”

Section: Lemma 4 ℎ(3) ≥mentioning

confidence: 99%

“…In 2006, the computer solution for = 6 was presented by Szekeres and Peters [5]; that, (6) = 17. In 2001, Avis et al [6] posed an interior point problem: for any integer ≥ 1, determine the smallest positive integer ( ) such that any finite point set of at least ( ) points has a subset for which the interior of the convex hull of the set contains exactly points in the set . Moreover, they also showed the results that (1) = 1 and (2) = 4.…”

Section: Introductionmentioning

confidence: 99%

“…In 2001, Avis et al [6] proved that 3 is the smallest positive integer such that any finite point set of at least 3 interior points has a subset for which the interior of the convex hull of the set contains exactly 3 or 4 points in the set . Moreover, they [11] also proved that 7 is the smallest positive integer such that any finite point set of at least 7 interior points has a subset for which the interior of the convex hull of the set contains exactly 4 or 5 points in the set .…”

Section: Introductionmentioning

confidence: 99%

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On the Existence of a Point Subset with 3 or 6 Interior Points

Sroysang

2013

ISRN Geometry

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For any finite planar point set in general position, an interior point of the set is a point of the set such that it is not on the boundary of the convex hull of the set . For any positive integer ≥ 3, let ℎ( ) be the smallest integer such that every finite planar point set with no three collinear points and with at least ℎ( ) interior points has a subset for which the interior of the convex hull of the set contains exactly or + 3 interior points of the set . In this paper, we prove that ℎ(3) = 8.

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Section: Conclusion and Discussionmentioning

confidence: 99%

“…This suffices to show the existence of a deficient point set of type (3,7,3,6). We construct a deficient point set of type (3,7,3,6) as shown in Figure 1. Hence, ℎ(3) ≥ 8.…”

Section: Lemma 4 ℎ(3) ≥mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Existence of a Point Subset with 3 or 6 Interior Points

Sroysang

2013

ISRN Geometry

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“…For any integer k ≥ 1, let g(k) be the smallest number such that every set of points P in general position in the plane, which contains at least g(k) interior points has a subset whose convex hull contains exactly k points of P in its interior. Avis, Hosono, and Urabe [2] determined that g(1) = 1, g(2) = 4 and g(3) ≥ 8. It is not known if g(k) exists for k ≥ 3.…”

Section: Open Problem 13 Is It True For Any Fixed Value Of K Thatmentioning

confidence: 99%

Problems and Results around the Erdös-Szekeres Convex Polygon Theorem

Bárány

Károlyi

2001

Discrete and Computational Geometry

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Problems on Points in General Position

Research Problems in Discrete Geometry

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In this section, just as in the rest of Chapter 8, all point sets are assumed to be in general position, that is, in the plane, they do not contain three points on a line, or, in d-dimensional space, d + 1 points in a hyperplane. These point sets can be described by their order types, as defined in Section 8.1.Perhaps the most famous problem concerning point sets in general position is the "Erdős-Szekeres convex polygon problem," also known as Esther Klein's problem or the Happy End problem* (see [MoS00], [BáK01], for recent surveys). What is the smallest number f ES (r) such that any set of f ES (r) points in general position in the plane contains r points that form the vertex set of a convex r-gon? Instead of saying that a point set contains the "vertex set of a convex r-gon," in the sequel we simply say that it contains a "convex r-gon."In their first joint paper, Erdős and Szekeres [ErS35] established the existence of this number by reducing the result to a Ramsey-type statement and rediscovering Ramsey's theorem for its proof. In a much later paper [ErS60], the same authors obtained the lower bound f ES (r) ≥ 2 r−2 + 1. The only known exact values of this function are f ES (3) = 3, f ES (4) = 5, and f ES (5) = 9 [Bo74], [KaKS70]. It is conjectured that the lower bound is sharp for all r. Conjecture 1 (Erdős-Szekeres [ErS35]) Any set of 2 r−2 + 1 points in general position in the plane contains a convex r-gon.

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On the existence of a point subset with a specified number of interior points

Cited by 17 publications

References 0 publications

On the Existence of a Point Subset with 3 or 6 Interior Points

On the Existence of a Point Subset with 3 or 6 Interior Points

Problems and Results around the Erdös-Szekeres Convex Polygon Theorem

Problems on Points in General Position

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