J. D. Horton, Sets with no empty convex 7-gons, Canadian Math. Bull. 26 (1983), 482-484. 13  Home/Search   Document Not in Database   Summary   Related Articles   Check

....there a number h(m) for every integer m 3, such that every n set with n h(m) contains an m hole Obviously h(3) 3, and it is easy to see that h(4) 5. The 9 point configuration depicted on Figure 1 verifies that h(5) 10. Harborth [15] proved in 1978 that h(5) 10, and in 1983 Horton [16] showed that h(m) does not exist for m 7 by constructing arbitrarily large sets without a 7 hole. The existence of h(6) is a problem that still remains open. Although Erdos and Szekeres already mentioned the generalization of their original problem to higher dimensions, it is still far from ....

....have more than j 2 = 2 compatible empty monochromatic triangles. # 3.2 Colored Horton sets Results claiming the non existence of monochromatic m holes can be obtained by appropriate colorings of the so called Horton sets, which are examples of point sets without 7 holes. A Horton set [16, 25] is a set H of n points sorted by x coordinates: p 1 x p 2 x p 3 x . x p n , such that the odd points p 1 , p 3 , and the even points p 2 , p 4 , are Horton sets and such that any line through two even points (the upper set) leaves all odd points below and any line through ....

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J. D. Horton. Sets with no empty convex 7-gons. Canad. Math. Bull., 26:482-- 484, 1983.

....is asked by Erd os in [2] for the value of the smallest integer B(k) such that any set of B(k) points contains an empty convex subset of size k. Values for B(k) are known for all values of k except k = 6. In [2] it is shown that B(3) 4, and B(4) 5. In [4] it is shown that B(5) 10. Horton [5] gives a construction showing that B(7) is not nite, that is, there are arbitrarily many points with no empty convex 7 gons. Horton s construction for 16 points is shown in Figure 1. The value of B(6) is not known, and this remains a tantalizing long outstanding open problem. Some experimental ....

....n, since we can always triangulate its interior, the trivial upper bound of F (n) is n 2. Fig. 1. Horton s construction with 16 points, partitioned into four convex cells. In the next section we prove the following theorem. Theorem 1. F (n) 2 Upper and lower bounds As was shown in [5] there exists sets n = 2 with no empty convex hexagons giving the lower bound n=4. We obtain here the lower bound for any integer n. 2 =m cos(2 =m) Fig. 2. An example to illustrate the lower bound Lemma 1. F (n) Proof. We rst construct a set of n = 2m points, such that we have m ....

J. D. Horton. Sets with no empty convex 7-gons. Canad. Math. Bull., 26:482-484, 1983.

....we could also define F to be the smallest value such that any set of size F contains some . ErdSs[3] proposed the study of finding bounds on Fk. It is trivial to prove that F = 1, F2 = 2, Fs = 3 and F4 = 5. The following results are known: Theorem 1.1 (Harborth[J] F = 10. Theorem 1. 2 (Horton[5]) F7 = oe. In other words, for each n there eeists a set of n points without empty convee 7 gon. Clearly this result also holds for k 7. For k = 6 no exact bounds on Ft are known. In [1] Avis and Rappaport give a method to determine whether a given set of points does contain an empty convex ....

Horton, J.D., Sets with no empty convex 7-gons, C. Math. Bull. 26 (1983) 482- 484. 9

....and every set of ten points contains an emptyconvex pentagon# our techniques can be applied to these cases. However, it is open whether there is a largest set with no emptyconvex hexagon (see, for example, 31] and there are arbitrarily large point sets that contain no emptyconvex heptagons [26]. Weknow of no similar results in higher dimensions. Our results suggest several open problems. None of our results is known to be optimal. Faster algorithms, or matching lower bounds, would be interesting. In particular, is it possible to find higher dimensional k point sets with minimum ....

J. D. Horton. Sets with no emptyconvex 7-gons. Canad. Math. Bull., 26:482--484, 1983.

....that for every k there is some n = f(k) 2 O(k) so that if n points are given in general position, a subset of k points can be found forming the vertices of a convex k gon. This does not work for empty convex polygons: arbitrarily large point sets are known that do not contain an empty 7 gon [13, 14]. This naturally raises the question whether such a k gon can be computed e#ciently; several papers study this problem [3, 6, 14] Because of Ramsey theory, finding a convex k gon takes time depending on k but not on n; therefore it can be solved in constant time for fixed k. The best known ....

....Unfortunately our methods do not su#ce to solve the third problem treated by Eppstein et al. finding a minimum area empty convex k gon, except for the special cases k = 4 and k = 5. This happens because of the lack of an appropriate Ramsey theorem, due to the counterexamples described by Horton [13]. 2 Nearest vertical neighbors We begin with the problem of nearest vertical neighbors for points and line segments; we use this as a subroutine in our minimum k gon algorithm. Given a point x and a non vertical line l, the vertical distance d(x, l) is simply the length of a vertical line ....

J.D. Horton. Sets with no empty convex 7-gons. Canad. Math. Bull. 26 (1983) 482--484. 9

....was considered in [E75] Let g(n) be the smallest integer so that if there are g(n) points in the plane , no three on a line, then there are always n of them which form the vertices of an empty convex n gon. It has been proved that g(n) exists for n = 3; 4; 5 [H78] does not exist for n 7 [H83], and its existence for n = 6 is still open. Two other related problems are concerned with the estimation of the number of convex k gons and the number of empty convex k gons determined by a set of n points. This last problem will be the subject of our paper. We call a set S, jSj 3) of points ....

....= ng. Similarly, denote by g k (S) the number of empty convex k gons determined by a set S of points and let g k (n) minfg k (S) S R 2 independent ; jSj = ng. Katchalski and Meir [KM88] proved that there is a constant K 200 such that for any n 3 n 1 2 g 3 (n) Kn 2 Horton [H83] constructed arbitrarily large sets of points H n ; for n = 2 t ; t 2 N having no empty convex 7 gons. This gives g k (n) 0 for k 7 . B ar any and F uredi [BF87] proved that n 2 O(n log n) g 3 (n) g 3 (H n ) 2n 2 1 4 n 2 O(n log n) g 4 (n) g 4 (H n ) 3n 2 b n 10 c g ....

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J. Horton, Sets with no empty convex 7-gons, Canadian Mathematical Bulletin, 26 (1983), 482-484.

....the maximum number g(k) of points in general position in the plane so that no k points form a vertex set of an empty convex polygon, i.e. a convex polygon whose interior is disjoint from the point set. It is easy to see that g(3) 2 and g(4) 4. Harborth [4] proved that g(5) 9, and Horton [5] showed that g(k) is infinite for k 7. It is a challenging open problem to decide whether g(6) is finite. Let g k (n) denote the minimum number of empty convex k gons induced by the k tuples of a set of n points in general position in the plane. B ar any and Furedi [1] proved that g 3 (n) n 2 ....

Horton, J. D.: Sets with no empty convex 7-gons. Canad. Math. Bull. 26 (1983), 482-484.

....and n; see, for example, 20] for recent developments. Other issues have been considered, such as the existence and computation ( 7] of large empty convex subsets (i.e. with no points of S interior to their hull) this is related to the convex partition number, 2 (S) It was shown by Horton [14] that there are sets with no empty convex chain larger than 6, so this implies that 2 (n) n=6. Tighter worst case bounds were given by Urabe [21, 22] Another possibility is to consider a simple polygon having a given set of vertices, that is as convex as possible . This has been studied in ....

J. Horton. Sets with no empty convex 7-gons. Canad. Math. Bull., 26, 482--484, 1983.

..... Erdos [3] proposed the study of finding bounds on . See [6] for an overview of the state of the art for this problem and related problems. It is trivial to prove that at 39 , 43139 , 47243 and nd 53 . The following results are known: Theorem 1.1 (Harborth[4] 21032 . Theorem 1. 2 (Horton[5]) 1 . In other words, for any size there exists a set of points without an empty convex 7 gon. Clearly this last result also holds for ) For , no exact bounds on are known. In [1] Avis and Rappaport give a method to determine whether a given set of points does ....

Horton, J.D., Sets with no empty convex 7-gons, C. Math. Bull. 26 (1983), pp. 482--484. 4

....and n; see, for example, 20] for recent developments. Other issues have been considered, such as the existence and computation ( 7] of large empty convex subsets (i.e. with no points of S interior to their hull) this is related to the convex partition number, 2 (S) It was shown by Horton [14] that there are sets with no empty convex chain larger than 6, so this implies that 2 (n) n=6. Tighter worstcase bounds were given by Urabe [21, 22] Another possibility is to consider a simple polygon having a given set of vertices, that is as convex as possible . This has been studied in ....

J. Horton. Sets with no empty convex 7-gons. Canad. Math. Bull., 26, 482--484, 1983.

....based on two di erent constructions. After we introduce some notation in Section 2, the rst proof is presented in Section 3. The second proof is based on the notion of so called d Horton sets which generalize Horton s construction of planar point sets that do not contain empty convex 7 gons, see [5, 10]. This notion is explained in Section 4 and is used in Section 5 for the second proof of Theorem 2. 2 Preliminaries Fix the dimension d 2. Identify, for every 1 e d, R e with the unique e dimensional subspace of R d spanned by the rst e coordinate axes. This way R f is identi ed ....

J.D. Horton, Sets with no empty convex 7-gons, Canadian Math. Bull. 26 (1983) 482-484.

....in the plane, many papers have studied problems of determining subsets of points in P that form polygons with particular properties. One such problem deals with finding empty convex k gons in a set of points (i.e. polygons that contain no points of P other than the k vertices) It is well known [12] that such k gons might not exist for k # 7. Algorithms to find such k gons have been presented in [4, 7, 14] The best known result works for arbitrary k in time O(T (n) where T (n) is the number of empty triangles in the set, which varies between O(n 2 ) and O(n 3 ) 7] Boyce, Dobkin, ....

J.D. Horton, Sets with no empty convex 7-gons, Canad. Math. Bull. 26 (1983), 482--484.

....and n; see, for example, 16] for recent developments. Other issues have been considered, such as the existence and computation ( 6] of large empty convex subsets (i.e. with no points of S interior to their hull) this is related to the convex partition number, 2 (S) It was shown by Horton [13] that there are sets with no empty convex chain larger than 6, so this implies that 2 (n) n=6. Tighter worst case bounds were given by Urabe [17, 18] Another possibility is to consider a simple polygon having a given set of vertices, that is as convex as possible . This has been studied in ....

J. Horton. Sets with no empty convex 7-gons. Canad. Math. Bull., 26:482--484, 1983.

....which explains the usefulness of such a classification scheme: Let H(n) be the smallest natural number, such that every set of H(n) points in the plane (no three of them collinear) contains the vertices of an empty convex n gon. It is known that H(5) 10 [Ha] and that H(7) does not exist [Ho]. Whether H(6) exists, is still an open problem. Suppose you want to attack this question algorithmically and test each configuration of a given number of points whether it contains an empty convex hexagon. Besides from the immense complexity for large configurations, how do you generate all ....

J.D.HORTON, Sets with no empty convex 7-gon, Canad. Math. Bull 26 (1983), 482-484

....and every set of ten points contains an empty convex pentagon; our techniques can be applied to these cases. However, it is open whether there is a largest set with no empty convex hexagon (see, for example, 31] and there are arbitrarily large point sets that contain no empty convex heptagons [26]. We know of no similar results in higher dimensions. Our results suggest several open problems. None of our results is known to be optimal. Faster algorithms, or matching lower bounds, would be interesting. In particular, is it possible to find higher dimensional k point sets with minimum ....

J. D. Horton. Sets with no empty convex 7-gons. Canad. Math. Bull., 26:482--484, 1983.

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J. D. Horton, Sets with no empty convex 7-gons, Canadian Math. Bull. 26 (1983), 482-484. 13

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J. D. Horton, Sets with no empty convex 7-gons, Canadian Math. Bull. 26 (1983), 482-484.

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J. D. Horton. Sets with no empty convex 7-gons. Canad. Math. Bull., 26:482484, 1983.

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J. D. Horton, Sets with no empty convex 7-gons, Canadian Math. Bull. 26 (1983), 482-484.

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J. D. Horton, Sets with no empty convex 7-gons, Canadian Math. Bull. 26 (1983), 482-484.

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J. D. Horton. Sets with no empty convex #-gons. Canad. Math. Bull., 26:482484, 1983.

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