On the maximum number of unit distances among \(n\) points in dimension four. (English) Zbl 0885.52015
Bárány, Imre (ed.) et al., Intuitive geometry. Proceedings of the 5th conference, Budapest, Hungary, September 3–8, 1995. Budapest: János Bolyai Mathematical Society. Bolyai Soc. Math. Stud. 6, 277-290 (1997).
The problem of determining the maximum number of distances among \(n\) points in the plane is a well-known problem by P. Erdős. [See P. Erdös, Am. Math. Mon. 53, 248-250 (1946; Zbl 0060.34805), A. Baker (ed.), B. Bollobás (ed.) and András Hajnal (ed.), ‘A tribute to Paul Erdős’ (1990; Zbl 0706.00007) and P. Erdős, A tribute to Paul Erdős, 467-478 (1990; Zbl 0709.11003) and Discrete geometry and convexity 440, 1-11 (1985; Zbl 0568.51011).]
The aim of this paper is to determine the exact maximum number of unit distances for dimension 4. The author shows the following
Theorem: The maximum number of unit distances among \(n\geq 5\) points in \(\mathbb{R}^4\) is \([1/4\cdot n^2]+ n\) if \(\left({\sin\nu_1 \pi\over[{1\over 2}n]}\right)^{-2}+ \left({\sin\nu_2\pi\over [{1\over 2}n]}\right)^{-2}= 4\) has a solution \(\nu_1,\nu_2\in \mathbb{N}\) with \(\nu_1,\nu_2< [n/2]\), and \([1/4\cdot n^2]+ n-1\) else.
Such a solution exists at least in the classical case \(8|n\), in which we can take \(\nu_1= \nu_2= n/8\).
For the proof of the Main Theorem several remarkable lemmas are given.
For the entire collection see [Zbl 0868.00054].
The aim of this paper is to determine the exact maximum number of unit distances for dimension 4. The author shows the following
Theorem: The maximum number of unit distances among \(n\geq 5\) points in \(\mathbb{R}^4\) is \([1/4\cdot n^2]+ n\) if \(\left({\sin\nu_1 \pi\over[{1\over 2}n]}\right)^{-2}+ \left({\sin\nu_2\pi\over [{1\over 2}n]}\right)^{-2}= 4\) has a solution \(\nu_1,\nu_2\in \mathbb{N}\) with \(\nu_1,\nu_2< [n/2]\), and \([1/4\cdot n^2]+ n-1\) else.
Such a solution exists at least in the classical case \(8|n\), in which we can take \(\nu_1= \nu_2= n/8\).
For the proof of the Main Theorem several remarkable lemmas are given.
For the entire collection see [Zbl 0868.00054].
Reviewer: E.Quaisser (Potsdam)
MSC:
| 52C10 | Erdős problems and related topics of discrete geometry |
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
| 51M04 | Elementary problems in Euclidean geometries |
