Distinct Distances from Three Points

Distinct Distances from Three Points

Let p1, p2, p3 be three noncollinear points in the plane, and let P be a set of n other points in the plane. We show that the number of distinct distances between p1, p2, p3 and the points of P is Ω(n), improving the lower bound Ω(n) of Elekes and Szabó [4] (and considerably simplifying the analysis).

Distinct distances between points and lines

We show that for m points and n lines in R2, the number of distinct distances between the points and the lines is Ω(m1/5n3/5), as long as m1/2 ≤ n ≤ m2. We also prove that for any m points in the plane, not all on a line, the number of distances between these points and the lines that they span is Ω(m4/3). The problem of bounding the number of distinct point-line distances can be reduced to the...

Three Distinct Distances in the Plane

On distinct sums and distinct distances

The paper (Discrete Comput. Geom. 25 (2001) 629) of Solymosi and Tóth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all ð 2Þn sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves that the number of distinct sums is at least ns ; where ds 1⁄4 1=cJs=2n is defined in...

Extended Distances Between Fuzzy Points

In this paper, we tackle the extension of aggregations of distances in R to the fuzzy numbers and fuzzy points and we investigate the fullfilment of the conditions of a fuzzy metric.