Cited By
View all- Houle MImai HImai KRobert JYamamoto P(1993)Orthogonal weighted linear L1 and L∞ approximation and applicationsDiscrete Applied Mathematics10.1016/0166-218X(93)90113-343:3(217-232)Online publication date: 10-Jun-1993
Points and triangles in the plane and halving planes in space
Authors: 
Boris Aronov, Bernard Chazelle, Herbert Edelsbrunner, Leonidas J. Guibas, Micha Sharir, Rephael WengerAuthors Info & Claims
Boris Aronov, Bernard Chazelle, Herbert Edelsbrunner, Leonidas J. Guibas, Micha Sharir, Rephael WengerAuthors Info & ClaimsPages 112 - 115
Abstract
We prove that for any set S of n points in the plane and n3-α triangles spanned by the points of S there exists a point (not necessarily of S) contained in at least n3-3α/(512 log5 n) of the triangles. This implies that any set of n points in three-dimensional space defines at most 6.4n8/3 log5/3 n halving planes.
References
[1]
N. Alan and E. Gy6ri. The number of small semispaces of a finite set of points in the plane. J. Combin. Theory Set. A 41 (1986), 154-157.
[2]
F. Aurenhammer. Power diagrams: properties, algorithms, and applications. SIAM J. Uompot. 16 (1987), 78-96.
[3]
I. B~r~ny. A generalization of Garathfiodory's theorem. Discrete Math. 40 (1982), 141-152.
[4]
I. B~r~ny, Z. Ffredi and L. Lov~sz. On the number of halving planes. In "Proc. 5th Ann. Sympos. Comput. Geom., 1989", 140-144.
[5]
}~. Boros and Z. Ffiredi. The number of triangles.covering the center of an n-set. Geometriae Dedicata 17 (1984), 69-77.
[6]
B. Chazelle, H. Edelsbrunner, L. J. Guibas, J. Hershberger, R. Seidel and M. Sharir. Selecting multiply covered points and reducing the size of Delaunay triangulations. In "Proc. 6th Ann. Sympos. Comput. Geom., 1990".
[7]
K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, II. Discrete Comput. Geom. 4 (1989), 387'-421.
[8]
H. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer-Vet|at, Heidelberg, Germany, 1987.
[9]
H. Edelsbrunner and E. Welzl. On the number of line separations of a finite set in the plane. J. Combin. Theory Set. A 38 (1985), 15-29.
[10]
P. ErdSs, L. Lov~sz, A. Simmons and E. G. Strauss. Dissection graphs of planar point sets. In A Surve~t of Combinatorial Theory, J. N. Srivastava et al., eds., 1973, 139-149, North-Holland, Amsterdam.
[11]
J. E. Goodman and R. Pollack. On the number of ksubsets of a set of n points in the plane. J. Combin. Theory, Set. A 36 (1984), 101-104.
[12]
L. LovLsz. On the number of halving lines. Ann. Univ. Sci. Budapest, E6tvSs, Sect. Math. 14 (1971), 107-108.
[13]
J. Pach, W. Steiger and E. Szemer~di. An upper bound on the number of planar k-sets. In "Proc. 30th Ann. IEEE Sympos. Found. Comput. Sci. 1989", 72-79.
Index Terms
- Points and triangles in the plane and halving planes in space
Recommendations
Points and triangles in the plane and halving planes in space
We prove that for any setS ofn points in the plane andn3 triangles spanned by the points inS there exists a point (not necessarily inS) contained in at leastn3 3 /(c log5n) of the triangles. This implies that any set ofn points in three-dimensional ...
Points and triangles in the plane and halving planes in space
We prove that for any setS ofn points in the plane andn3 triangles spanned by the points inS there exists a point (not necessarily inS) contained in at leastn3 3 /(c log5n) of the triangles. This implies that any set ofn points in three-dimensional ...
Triangles in arrangements of points and lines in the plane
A conjecture of de Caen and Szekely [D. de Caen, L.A. Szekely, J. Combin. Theory Ser. A 77 (1997) 268-278] asserts that every arrangement of n points and m lines in the plane contains at most O(nm) triangles with vertices in the points and sides on the ...
Comments
Information & Contributors
Information
Published In
Copyright © 1990 ACM.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]
Sponsors
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 01 May 1990
Check for updates
Qualifiers
- Article
Conference
Acceptance Rates
Overall Acceptance Rate 625 of 1,685 submissions, 37%
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- View Citations1Total Citations
- 52Total Downloads
- Downloads (Last 12 months)18
- Downloads (Last 6 weeks)2
Reflects downloads up to 26 Jul 2024
Other Metrics
Citations
Cited By
View all- Houle MImai HImai KRobert JYamamoto P(1993)Orthogonal weighted linear L1 and L∞ approximation and applicationsDiscrete Applied Mathematics10.1016/0166-218X(93)90113-343:3(217-232)Online publication date: 10-Jun-1993
View Options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in