Pach, János and Swanepoel, Konrad J. (2015) Double-normal pairs in space. Mathematika, 61 (1). pp. 259-272. ISSN 0025-5793
Abstract
A double-normal pair of a finite set $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$S of points that spans $\mathbb{R}^d$Rd is a pair of points $\{\boldsymbol {p},\boldsymbol {q} \}${p,q} from $S$S such that $S$S lies in the closed strip bounded by the hyperplanes through $\boldsymbol {p}$p and $\boldsymbol {q}$q perpendicular to $\boldsymbol {p}\boldsymbol {q}$pq. A double-normal pair $\{\boldsymbol {p},\boldsymbol {q} \}${p,q} is strict if$S\setminus \{\boldsymbol {p},\boldsymbol {q} \}$S∖{p,q} lies in the open strip. The problem of estimating the maximum number $N_d(n)$Nd(n) of double-normal pairs in a set of $n$n points in $\mathbb{R}^d$Rd, was initiated by Martini and Soltan [Discrete Math. 290 (2005), 221–228]. It was shown in a companion paper that in the plane, this maximum is $3\lfloor n/2\rfloor $3⌊n/2⌋, for every $n>2$n>2. For $d\geq 3$d⩾3, it follows from the Erdős–Stone theorem in extremal graph theory that $N_d(n)=\frac12(1-1/k)n^2 + o(n^2)$Nd(n)=12(1−1/k)n2+o(n2) for a suitable positive integer $k=k(d)$k=k(d). Here we prove that $k(3)=2$k(3)=2 and, in general, $\lceil d/2\rceil \leq k(d)\leq d-1$⌈d/2⌉⩽k(d)⩽d−1. Moreover, asymptotically we have $\lim _{n\rightarrow \infty }k(d)/d=1$limn→∞k(d)/d=1. The same bounds hold for the maximum number of strict double-normal pairs.
| Item Type: | Article |
|---|---|
| Official URL: | http://journals.cambridge.org/ |
| Additional Information: | © 2015 University College London |
| Library of Congress subject classification: | Q Science > QA Mathematics |
| Sets: | Departments > Management |
| Projects: | 200021-137574, 200020-144531, OTKA NN 102029 under the EuroGIGA programs ComPoSe and GraDR, CCF-08-30272 |
| Funders: | Swiss National Science Foundation, Swiss National Science Foundation, Hungarian Science Foundation, National Science Foundation (NSF) |
| Date Deposited: | 05 Sep 2014 08:38 |
| URL: | http://eprints.lse.ac.uk/59275/ |
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