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An incidence theorem in higher dimensions
 Discrete Comput. Geom
"... Abstract. We prove almost tight bounds on incidences between points and kdimensional varieties of bounded degree in R d. Our main tools are the Polynomial Ham Sandwich Theorem and induction on both the dimension and the number of points. ..."
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Abstract. We prove almost tight bounds on incidences between points and kdimensional varieties of bounded degree in R d. Our main tools are the Polynomial Ham Sandwich Theorem and induction on both the dimension and the number of points.
On a question of Bourgain about geometric incidences
 Combinat. Probab. Comput
"... Given a set of s points and a set of n 2 lines in threedimensional Euclidean space such that each line is incident to n points but no n lines are coplanar, then we have s = Ω(n 11/4). This is the first nontrivial answer to a question recently posed by Jean Bourgain. 1 ..."
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Given a set of s points and a set of n 2 lines in threedimensional Euclidean space such that each line is incident to n points but no n lines are coplanar, then we have s = Ω(n 11/4). This is the first nontrivial answer to a question recently posed by Jean Bourgain. 1
On the number of tetrahedra with minimum, unit, and distinct volumes in threespace ∗
, 710
"... We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by n points in 3space, and in general in d dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by n points in R3 is at most 2 3n3 − O(n2), and there are point sets for ..."
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We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by n points in 3space, and in general in d dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by n points in R3 is at most 2 3n3 − O(n2), and there are point sets for which this number is 3 16n3 − O(n2). We also present an O(n3) time algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby extend an algorithm of Edelsbrunner, O’Rourke, and Seidel. In general, for every k, d ∈ N, 1 ≤ k ≤ d, the maximum number of kdimensional simplices of minimum (nonzero) volume spanned by n points in R d is Θ(n k). (ii) The number of unitvolume tetrahedra determined by n points in R 3 is O(n 7/2), and there are point sets for which this number is Ω(n 3 log log n). (iii) For every d ∈ N, the minimum number of distinct volumes of all fulldimensional simplices determined by n points in R d, not all on a hyperplane, is Θ(n). 1
On the number of tetrahedra with minimum,unit, and distinct volumes in threespace
"... Abstract We formulate and give partial answers to several combinatorial problems on fourtuples of n pointsin threespace. (i) The number of minimum (nonzero) volume tetrahedra spanned by n points in R3 is \Theta (n3). (ii) The number of unitvolume tetrahedra determined by n points in R3 is O(n7=2) ..."
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Abstract We formulate and give partial answers to several combinatorial problems on fourtuples of n pointsin threespace. (i) The number of minimum (nonzero) volume tetrahedra spanned by n points in R3 is \Theta (n3). (ii) The number of unitvolume tetrahedra determined by n points in R3 is O(n7=2), and thereare point sets for which this number is \Omega (n3 log log n). (iii) The tetrahedra determined by n pointsin R3, not all on a plane, have at least \Omega (n) distinct volumes, and there are point sets for which thisnumber is O(n); this gives a first partial answer for the threedimensional case to an old question ofErd&quot;&quot;os, Purdy, and Straus. We also give an O(n3) time algorithm for reporting all tetrahedra of minimumnonzero volume, and thereby extend an early algorithm of Edelsbrunner, O'Rourke, and Seidel. 1 Introduction Typical Erd&quot;&quot;os type problems in extremal discrete mathematics ask for the minimum or maximum numberof certain configurations over all inputs of a given size. They are easy to formulate but often extremely hard to answer. Their impact on mathematics and computer science has been enormous, not only becauseof specific algorithms based on combinatorial bounds but also because they have triggered the development of theoretical and practical methods that turned out to be applicable elsewhere.Some of the most simply formulated yet notoriously hard Erd&quot;&quot;os type problems occur in combinatorial geometry. In 1946, Erd&quot;&quot;os [21] asked two questions on distances: (1) at most how many times can agiven distance occur among n points in the plane; (2) what is the minimum number of distinct distancesdetermined by n points in the plane? The difficulty of the, so called, unit distance and distinct distanceproblems is still to be measured. Erd&quot;&quot;os and Purdy [22, 23] generalized the unit and distinct distance