# The number of double-normals in space

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## Abstract

Given a set *V* of points in \({\mathbb {R}}^d\), two points *p*, *q* from *V* form a double-normal pair, if the set *V* lies between two parallel hyperplanes that pass through *p* and *q*, respectively, and that are orthogonal to the segment *pq*. In this paper we study the maximum number \(N_d(n)\) of double-normal pairs in a set of *n* points in \({\mathbb {R}}^d\). It is not difficult to get from the famous Erdős–Stone theorem that \(N_d(n) = \frac{1}{2}(1-1/k)n^2+o(n^2)\) for a suitable integer \(k = k(d)\) and it was shown in a paper by Pach and Swanepoel that \(\lceil d/2\rceil \le k(d)\le d-1\) and that asymptotically \(k(d)\gtrsim d-O(\log d)\). In this paper we sharpen the upper bound on *k*(*d*), which, in particular, gives \(k(4)=2\) and \(k(5)=3\) in addition to the equality \(k(3)=2\) established by Pach and Swanepoel. Asymptotically we get \(k(d)\le d- \log _2k(d) = d - (1+ o(1)) \log _2k(d)\) and show that this problem is connected with the problem of determining the maximum number of points in \({\mathbb {R}}^d\) that form only acute (or non-obtuse) angles.

## Keywords

Double-normal pairs Acute angles Number of edges Geometric graphs## Notes

### Acknowledgments

The author is very grateful to János Pach, Sasha Polyanskii and Konrad Swanepoel for several stimulating discussions on the subject and to one of the reviewers, who had very carefully read the manuscript and pointed out several problems with the proofs and the exposition. It helped to improve a lot the presentation of the paper. Research was supported in part by the Swiss National Science Foundation Grants 200021-137574 and 200020-14453 and by the Grant N 15-01-03530 of the Russian Foundation for Basic Research.

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