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97
On the Number of Congruent Simplices in a Point Set
 DISCRETE & COMPUTATIONAL GEOMETRY
, 2002
"... We derive improved bounds on the number of kdimensional simplices spanned by a set of n points in R(sup d) that are congruent to a given ksimplex, for k < or = d  1. Let f(sup d)(sub K)(n) be the maximum number of ksimplices spanned by a set of n points in R(sup d) that are congruent to a given ..."
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Cited by 12 (2 self)
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We derive improved bounds on the number of kdimensional simplices spanned by a set of n points in R(sup d) that are congruent to a given ksimplex, for k < or = d  1. Let f(sup d)(sub K)(n) be the maximum number of ksimplices spanned by a set of n points in R(sup d) that are congruent to a given ksimplex. We prove that f(sup 3)(sub 2) (n) = O(n sup 5/3) times 2 sup O(alpha (sup 2)(n)), f(sup 4)(sub 2)(n) = O(n (sup 2) + epsilon), f(sup 5)(sub 2) (n)) = theta (n (sup 7/3), and f(sup 4)(sub 3)(n) = O(n (sup 9/4) plus epsilon). We also derive a recurrence to bound f(sup d)(sub k) (n) for arbitrary values of k and d, and use it to derive the bound f(sup d)(sub k(n) = O(n(sup d/2)) for d < or = 7 and k < or = d  2. Following Erdo's and Purdy, we conjecture that this bound holds for larger values of d as well, and for k < or = d  2.
Reconstructing Sets From Interpoint Distances
 of Algorithms Combin
, 2002
"... Which point sets realize a given distance multiset? Interesting cases include the "turnpike problem" where the points lie on a line, the "beltway problem" where the points lie on a loop, and multidimensional versions. We are interested both in the algorithmic problem of determining such point sets f ..."
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Cited by 11 (0 self)
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Which point sets realize a given distance multiset? Interesting cases include the "turnpike problem" where the points lie on a line, the "beltway problem" where the points lie on a loop, and multidimensional versions. We are interested both in the algorithmic problem of determining such point sets for a given collection of distances and the combinatorial problem of finding bounds on the maximum number of different solutions. These problems have applications in genetics and crystallography.
Distinct distances in three and higher dimensions
 Combin. Probab. Comput
, 2003
"... Improving an old result of Clarkson et al., we show that the number of distinct distances determined by a set P of n points in threedimensional space is Ω(n77/141−ε) = Ω(n0.546), for any ε> 0. Moreover, there always exists a point p ∈ P from which there are at least so many distinct distances to th ..."
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Cited by 11 (2 self)
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Improving an old result of Clarkson et al., we show that the number of distinct distances determined by a set P of n points in threedimensional space is Ω(n77/141−ε) = Ω(n0.546), for any ε> 0. Moreover, there always exists a point p ∈ P from which there are at least so many distinct distances to the remaining elements of P. The same result holds for points on the threedimensional sphere. As a consequence, we obtain analogous results in higher dimensions. 1
Incidences in Three Dimensions and Distinct Distances in the Plane (Extended Abstract)
, 2010
"... We first describe a reduction from the problem of lowerbounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new setup, but ..."
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Cited by 11 (5 self)
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We first describe a reduction from the problem of lowerbounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new setup, but are still unable to fully resolve them. Instead, we adapt the recent new algebraic analysis technique of Guth and Katz [9], as further developed by Elekes et al. [6], to obtain sharp bounds on the number of incidences between these helices or parabolas and points in R 3. Applying these bounds, we obtain, among several other results, the upper bound O(s 3) on the number of rotations (rigid motions) which map (at least) three points of S to three other points of S. In fact, we show that the number of such rotations which map at least k ≥ 3 points of S to k other points of S is close to O(s 3 /k 12/7). One of our unresolved conjectures is that this number is O(s 3 /k 2), for k ≥ 2. If true, it would imply the lower bound Ω(s / log s) on the number of distinct distances in the plane.
Fourier bases and a distance problem of Erdős
 Amer. J. Math
, 1999
"... Abstract. We prove that no ball admits a nonharmonic orthogonal basis of exponentials. We use a combinatorial result, originally studied by Erdős, which says that the number of distances determined by n points in Rd is at least Cdn 1 d +ǫd, ǫd> 0. Introduction and statement of results Fourier bases ..."
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Cited by 10 (2 self)
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Abstract. We prove that no ball admits a nonharmonic orthogonal basis of exponentials. We use a combinatorial result, originally studied by Erdős, which says that the number of distances determined by n points in Rd is at least Cdn 1 d +ǫd, ǫd> 0. Introduction and statement of results Fourier bases. Let D be a domain in R d, i.e., D is a Lebesgue measurable subset of R d with finite nonzero Lebesgue measure. We say that D is a spectral set if L 2 (D) has orthogonal basis of the form EΛ = {e 2πix·λ}
Distance sets of welldistributed planar sets, preprint
, 2002
"... Abstract. Let X be a 2dimensional normed space, and let BX be the unit ball in X. We discuss the question of how large the set of extremal points of BX may be if X contains a welldistributed set whose distance set ∆ satisfies the estimate  ∆ ∩ [0, N]  ≤ CN 3/2−ǫ. We also give a necessary and s ..."
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Cited by 9 (5 self)
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Abstract. Let X be a 2dimensional normed space, and let BX be the unit ball in X. We discuss the question of how large the set of extremal points of BX may be if X contains a welldistributed set whose distance set ∆ satisfies the estimate  ∆ ∩ [0, N]  ≤ CN 3/2−ǫ. We also give a necessary and sufficient condition for the existence of a welldistributed set with  ∆ ∩ [0, N]  ≤ CN.
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.
Isosceles Triangles Determined By a Planar Point Set
"... It is proved that, for any " > 0 and n > n 0 ("), every set of n points in the plane has at most n 5e 1 + triples that induce isosceles triangles. (Here e denotes the base of the natural logarithm, so the exponent is roughly 2:136.) This easily implies the best currently known lower bound, n 5 ..."
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Cited by 7 (2 self)
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It is proved that, for any " > 0 and n > n 0 ("), every set of n points in the plane has at most n 5e 1 + triples that induce isosceles triangles. (Here e denotes the base of the natural logarithm, so the exponent is roughly 2:136.) This easily implies the best currently known lower bound, n 5e 1 , for the smallest number of distinct distances determined by n points in the plane, due to Solymosi{C. Toth and Tardos.
Some scalable parallel algorithms for geometric problems
 Journal of Parallel and Distributed Computing
, 1999
"... This paper considers a variety of geometric pattern recognition problems on input sets of size n using a coarse grained multicomputer model consisting of p processors with 0(n p) local memory each (i.e., 0(n p) memory cells of 3(log n) bits apiece), where the processors are connected to an arbitrary ..."
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Cited by 7 (2 self)
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This paper considers a variety of geometric pattern recognition problems on input sets of size n using a coarse grained multicomputer model consisting of p processors with 0(n p) local memory each (i.e., 0(n p) memory cells of 3(log n) bits apiece), where the processors are connected to an arbitrary interconnection network. It introduces efficient scalable parallel algorithms for a number of geometric problems including the rectangle finding problem, the maximal equally spaced collinear points problem, and the point set pattern matching problem. All of the algorithms presented are scalable in that they are applicable and efficient over a very wide range of ratios of problem size to number of processors. In addition to the practicality imparted by scalability, these algorithms are easy to implement in that all required communications can be achieved by a small number of calls to standard global routing operations.