Can I read Distinct Distances Between a Collinear Set and an Arbitrary Set of Points on EtoBox?

Distinct Distances Between a Collinear Set and an Arbitrary Set of Points by Bruner, Ariel; Sharir, Micha is a scholarly article available to read on EtoBox.

What is Distinct Distances Between a Collinear Set and an Arbitrary Set of Points about?

We consider the number of distinct distances between two finite sets of points in ${\bf R}^k$, for any constant dimension $k\ge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary points, such that no hyperplane orthogonal to $l$ and no hypercylinder having $l$ as its axis contains more than $O(1)$ points of $P_2$. The number of distinct distances between $P_1$ and $P_2$ is then $$ \Omega\left(\min\left\{ n^{2/3}m^{2/3},\; \frac{n^{10/11}m^{4/11}}{\log^{2/11}m},\; n^2,\; m^2\right\}\right) . $$ Without the assumption on $P_2$, there exist sets $P_1$, $P_2$ as above, with only $O(m+n)$ distinct distances between them.

Author

Bruner, Ariel; Sharir, Micha

Published

2016

Language

EN