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12
Lenses in arrangements of pseudocircles and their applications
 J. ACM
, 2004
"... Abstract. A collection of simple closed Jordan curves in the plane is called a family of pseudocircles if any two of its members intersect at most twice. A closed curve composed of two subarcs of distinct Work on this article by P. Agarwal, J. Pach, and M. Sharir has been supported by a joint grant ..."
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Cited by 22 (9 self)
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Abstract. A collection of simple closed Jordan curves in the plane is called a family of pseudocircles if any two of its members intersect at most twice. A closed curve composed of two subarcs of distinct Work on this article by P. Agarwal, J. Pach, and M. Sharir has been supported by a joint grant from the U.S.–Israel Binational Science Foundation. Work by P. Agarwal has also been supported by NSF grants EIA9870724, EIA9972879, ITR333
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 9 (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.
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.
On Distinct Distances from a Vertex of a Convex Polygon
"... Given a set P of n points in convex position in the plane, we prove that there exists a point p 2 P such that the number of distinct distances from p is at least d(13n  6)/36e. The best previous bound, dn=3e, from 1952, is due to Leo Moser. ..."
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Cited by 3 (1 self)
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Given a set P of n points in convex position in the plane, we prove that there exists a point p 2 P such that the number of distinct distances from p is at least d(13n  6)/36e. The best previous bound, dn=3e, from 1952, is due to Leo Moser.
DISTINCT DISTANCES IN GRAPH DRAWINGS
, 2008
"... The distancenumber of a graph G is the minimum number of distinct edgelengths over all straightline drawings of G in the plane. This definition generalises many wellknown concepts in combinatorial geometry. We consider the distancenumber of trees, graphs with no K − 4minor, complete bipartite g ..."
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Cited by 1 (0 self)
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The distancenumber of a graph G is the minimum number of distinct edgelengths over all straightline drawings of G in the plane. This definition generalises many wellknown concepts in combinatorial geometry. We consider the distancenumber of trees, graphs with no K − 4minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distancenumber of graphs with bounded degree. We prove that nvertex graphs with bounded maximum degree and bounded treewidth have distancenumber in O(log n). To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth 2 and polynomial distancenumber. Similarly, we prove that there exist graphs with maximum degree 5 and arbitrarily large distancenumber. Moreover, as ∆ increases the existential lower bound on the distancenumber of ∆regular graphs tends to Ω(n0.864138). 1
The number of distinct distances from a vertex of a convex polygon
 J. Comput. Geom
"... Erdős conjectured in 1946 that every npoint set P in convex position in the plane has a point that determines at least ⌊n/2 ⌋ distinct distances to the other points of P. In 2006 Dumitrescu improved the best known lower bound for this problem, from n/3 to 13n/36 − O(1). A crucial step in his argume ..."
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Erdős conjectured in 1946 that every npoint set P in convex position in the plane has a point that determines at least ⌊n/2 ⌋ distinct distances to the other points of P. In 2006 Dumitrescu improved the best known lower bound for this problem, from n/3 to 13n/36 − O(1). A crucial step in his argument is showing that P must determine at most n2 (1 − 1/12) isosceles triangles. In this paper we show that Dumitrescu’s bound can be further improved, though our improvement is quite small. We show that the number of isosceles triangles determined by P is at most n2 (1 − 1/11.981), and we conclude that there exists a point of P that determines at least
On Distinct Distances and Incidences: Elekes’s Transformation and the New Algebraic Developments ∗
, 2010
"... We first present a transformation that Gyuri Elekes has devised, about a decade ago, from the celebrated problem of Erdős 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 parabo ..."
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We first present a transformation that Gyuri Elekes has devised, about a decade ago, from the celebrated problem of Erdős 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. Elekes has offered conjectures involving the new setup, which, if correct, would imply that the number of distinct distances in an selement point set in the plane is always Ω(s/log s). Unfortunately, these conjectures are still not fully resolved. We then review the recent progress made on the transformed incidence problem, based on a new algebraic approach, originally introduced by Guth and Katz. Full details of the results reviewed