Double-normal pairs in space

Abstract

A double-normal pair of a finite set $S$ of points from $R^d$ is a pair of points $\{p,q\}$ from $S$ such that $S$ lies in the closed strip bounded by the hyperplanes through $p$ and $q$ perpendicular to $pq$. A double-normal pair $pq$ is strict if $S\setminus \{p,q\}$ lies in the open strip. The problem of estimating the maximum number $N_d(n)$ of double-normal pairs in a set of $n$ points in $R^d$, was initiated by Martini and Soltan (2006). It was shown in a companion paper that in the plane, this maximum is $3\lfloor n/2\rfloor$, for every $n>2$. For $d\ge 3$, it follows from the Erdős-Stone theorem in extremal graph theory that $N_d(n)=\frac{1}{2}(1-1/k)n^2+o(n^2)$ for a suitable positive integer $k=k(d)$. Here we prove that $k(3)=2$ and, in general, $\lceil d/2\rceil \le k(d)\le d-1$. Moreover, asymptotically we have ${\lim }_{n\to \infty }k(d)/d=1$. The same bounds hold for the maximum number of strict double-normal pairs.

15 pages, 1 figure

References in corpus (2)

• Unit distances and diameters in Euclidean spaces
• Diameter graphs in ${\mathbb{R}}^4$

Cited by in corpus (1)

• Number of double-normal pairs in space