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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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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...
Notes on geometric graph theory
 Discrete and Computational Geometry: Papers from DIMACS special year, volume 6 of DIMACS series, 273–285, AMS
, 1991
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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 11 (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
 J. Combin. Theory Ser. A
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How many unit equilateral triangles can be generated by n points in convex position
 American Mathematical Monthly
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The maximum number of empty congruent triangles determined by a point set, Revue Roumaine de Math. Pures et Appliquées 50
, 2005
"... Let S be a set of n points in the plane and consider a family of (nondegenerate) pairwise congruent triangles whose vertices belong to S. While the number of such triangles can grow superlinearly in n — as it happens in lattice sections of the integer grid — it has been conjectured by Brass that the ..."
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Let S be a set of n points in the plane and consider a family of (nondegenerate) pairwise congruent triangles whose vertices belong to S. While the number of such triangles can grow superlinearly in n — 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 Ω(n logn) empty congruent triangles. 1