منابع مشابه
Unit Distances in Three Dimensions
We show that the number of unit distances determined by n points in R is O(n), slightly improving the bound of Clarkson et al. [5], established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft stage, a similar proof of our main result was posted to the arXiv by Joshua Zahl [25].
متن کاملBreaking the 3/2 barrier for unit distances in three dimensions
We prove that every set of n points in R3 spans O(n295/197+ε) unit distances. This is an improvement over the previous bound of O(n3/2). A key ingredient in the proof is a new result for cutting circles in R3 into pseudo-segments.
متن کاملLines Avoiding Unit Balls in Three Dimensions
Let B be a set of n unit balls in R3. We show that the combinatorial complexity of the space of lines in R3 that avoid all the balls of B is O(n3+ε), for any ε > 0. This result has connections to problems in visibility, ray shooting, motion planning, and geometric optimization.
متن کاملOn unit distances in a convex polygon
For any convex quadrilateral, the sum of the lengths of the diagonals is greater than the corresponding sum of a pair of opposite sides, and all four of its interior angles cannot be simultaneously acute. In this article, we use these two properties to estimate the number of unit distance edges in convex n-gons and we: (i) exhibit three large groups of cycles formed by unit distance edges that ...
متن کاملUnit Distances and Diameters in Euclidean Spaces
We show that the maximum number of unit distances or of diameters in a set of n points in d-dimensional Euclidean space is attained only by specific types of Lenz constructions, for all d ≥ 4 and n sufficiently large, depending on d. As a corollary we determine the exact maximum number of unit distances for all even d ≥ 6, and the exact maximum number of diameters for all d ≥ 4, for all n suffi...
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ژورنال
عنوان ژورنال: Combinatorics, Probability and Computing
سال: 2012
ISSN: 0963-5483,1469-2163
DOI: 10.1017/s0963548312000144