Unit distances

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].

On 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 ...

Unit 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...

Movement minimization for unit distances in conveyor flow shop processing

We consider the movement minimization problem in a conveyor flow shop processing controlled by one worker for all machines. A machine can only execute tasks if the worker is present. Each machine can serve as a bu¤er. The objective is to minimize the total number of movements of the worker for the processing of all jobs. Such minimization problems usually appear in conveyor controlled manufactu...

A k-server problem with parallel requests and unit distances