×

Subtended angles. (English) Zbl 1410.52016

In this paper, the authors study the following problem: Suppose that \(d\geq2\) and \(m\) are fixed. What is the largest \(n\) such that, given any \(n\) distinct angles \(0 < \theta_1, \theta_2, \dots, \theta_n < \pi\), we can realise all these angles by placing \(m\) points in \(\mathbb{R}^{d}\)?
The paper has six sections. The first section is an introductive chapter which presents the main results that are going to be demonstrated. In Section 2, the authors prove the lower bound in 2 dimensions, and in Section 3 the upper bound in 2 dimensions. In the next section, they prove the higher dimensional upper bound. In Section 5, the authors prove the following theorem: “Suppose that \(d\) and \(\epsilon\) are fixed. Then there exists a constant \(c\) such that any \(n = dm-c\) angles, all lying between \(\epsilon\) and \(\pi-\epsilon\), can be realised using \(m\) points.” In the last section, the authors propose some open questions.

MSC:

52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] P. Brass, W. Moser and J. Pach, Research Problems in Discrete Geometry, Springer, New York, 2005. · Zbl 1086.52001
[2] J. H. Conway, H. T. Croft, P. Erdős and M. J. T. Guy, On the distribution of values of angles determined by coplanar points, Journal of the London Mathematical Society (2) 19 (1979), 137-143. · Zbl 0414.05006 · doi:10.1112/jlms/s2-19.1.137
[3] L. Danzer and B. Grünbaum, Über zwei Probleme bezüglich konvexer Körper von P. Erdős und von V. L. Klee, Mathematische Zeitschrift 79 (1962), 95-99. · Zbl 0188.27602 · doi:10.1007/BF01193107
[4] P. Erdős, On sets of distances of n points, The American Mathematical Monthly 53 (1946), 248-250. · Zbl 0060.34805 · doi:10.2307/2305092
[5] P. Erdős, Advanced Problems and Solutions: Solutions: 4306, The American Mathematical Monthly 69 (1962), 173. · doi:10.2307/2312563
[6] P. Erdős and Z. Füredi, The greatest angle among n points in the d-dimensional Euclidean space, in Combinatorial Mathematics (Marseille-Luminy, 1981), North-HollandMathematics Studies, Vol. 75, North-Holland, Amsterdam, 1983, pp. 275-283. · Zbl 0534.52007
[7] Z. Füredi and G. Szigeti, Four points and four angles in the plane, submitted. · Zbl 0046.14101
[8] D.J.H. Garling, A course in galois theory, Cambridge University Press, 1986. · Zbl 0608.12025
[9] L. Guth and N. H. Katz, On the Erdős distinct distances problem in the plane, Annals of Mathematics (2) 181 (2015), 155-190. · Zbl 1310.52019 · doi:10.4007/annals.2015.181.1.2
[10] V. Harangi, Acute sets in Euclidean spaces, SIAM Journal on Discrete Mathematics 25 (2011), 1212-1229. · Zbl 1237.05207 · doi:10.1137/100808095
[11] H. Hopf and E. Pannwitz, Aufgabe nr. 167, Jahresbericht der Deutschen Mathematiker- Vereinigung 43 (1934), 114. · JFM 61.0651.03
[12] L. Moser, On the different distances determined by n points, The American Mathematical Monthly 59 (1952), 85-91. · Zbl 0046.14101 · doi:10.2307/2307105
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.