Subtended angles

Paul Balister, Zoltán Füredi, Béla Bollobás, Imre Leader, Mark Walters

Research output: Contribution to journalArticle

Abstract

Suppose that d ≥ 2 and m are fixed. For which n is it the case that any n angles can be realised by placing m points in Rd? A simple degrees of freedom argument shows that m points in R2 cannot realise more than 2m - 4 general angles. We give a construction to show that this bound is sharp when m ≥ 5. In d dimensions the degrees of freedom argument gives an upper bound of dm−(d+12) general angles. However, the above result does not generalise to this case; surprisingly, the bound of 2m-4 from two dimensions cannot be improved at all. Indeed, our main result is that there are sets of 2m - 3 angles that cannot be realised by m points in any dimension.

Original languageEnglish
Pages (from-to)995-1012
Number of pages18
JournalIsrael Journal of Mathematics
Volume214
Issue number2
DOIs
Publication statusPublished - Jul 1 2016

ASJC Scopus subject areas

  • Mathematics(all)

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  • Cite this

    Balister, P., Füredi, Z., Bollobás, B., Leader, I., & Walters, M. (2016). Subtended angles. Israel Journal of Mathematics, 214(2), 995-1012. https://doi.org/10.1007/s11856-016-1370-1