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Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 78 (22 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
Geometric Graph Theory
, 1999
"... A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turántype and Ramseytype extremal problems for geometric graphs, and discusses their generalizations and applications. ..."
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Cited by 13 (0 self)
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A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turántype and Ramseytype extremal problems for geometric graphs, and discusses their generalizations and applications.
TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Cited by 13 (0 self)
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
Geometric Representations of Graphs
 IN PAUL ERDÖS, PROC. CONF
, 1999
"... The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph theory. We survey the surprisingly large variety of graph properties related to geometric representations, c ..."
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Cited by 8 (0 self)
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The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph theory. We survey the surprisingly large variety of graph properties related to geometric representations, construction methods for geometric representations, and their applications in proofs and algorithms.
The maximum number of times the same distance can occur among the vertices of a convex ngon is O(n log n)
, 2001
"... We present a short proof of Füredi's theorem [F] stated in the title. ..."
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Cited by 6 (3 self)
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We present a short proof of Füredi's theorem [F] stated in the title.
THE MAXIMUM NUMBER OF EMPTY CONGRUENT TRIANGLES DETERMINED BY A POINT SET
, 2005
"... Let ¡ be a set of ¢ points in the plane and consider a family of (nondegenerate) pairwise congruent triangles whose vertices belong to ¡. While the number of such triangles can grow superlinearly in ¢ — as it happens in lattice sections of the integer grid — it has been conjectured by Brass that th ..."
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Cited by 1 (1 self)
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Let ¡ be a set of ¢ points in the plane and consider a family of (nondegenerate) pairwise congruent triangles whose vertices belong to ¡. While the number of such triangles can grow superlinearly in ¢ — as it happens in lattice sections of the integer grid — it has been conjectured by Brass that the number of pairwise congruent empty triangles is only at most linear. We disprove this conjecture by constructing point sets with£¥¤¦¢¨§�©���¢� � empty congruent triangles.
How many unit equilateral triangles can be generated by n points in convex position?
 Amer. Math. Monthly
"... What is the maximum number of times that the unit distance can occur among the distances between n points in the plane? This more than ftyyearold question of Paul Erdös, published in this MONTHLY [5], opened a whole new area of research in combinatorial geometry [10]. ..."
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Cited by 1 (1 self)
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What is the maximum number of times that the unit distance can occur among the distances between n points in the plane? This more than ftyyearold question of Paul Erdös, published in this MONTHLY [5], opened a whole new area of research in combinatorial geometry [10].
How Many Unit Equilateral Triangles Can Be Induced by n Points in Convex Position?
, 2001
"... Any set of n points in strictly convex position in the plane has at most b c triples that induce equilateral triangles of side length one. This bound cannot be improved. The case of general triangles is also discussed. ..."
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Any set of n points in strictly convex position in the plane has at most b c triples that induce equilateral triangles of side length one. This bound cannot be improved. The case of general triangles is also discussed.