Subtended angles

Authors

Paul Balister, University of MemphisFollow
Zoltán Füredi, Alfréd Rényi Institute of Mathematics
Béla Bollobás, University of MemphisFollow
Imre Leader, Faculty of MathematicsFollow
Mark Walters, Queen Mary University of LondonFollow

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.

Publication Title

Israel Journal of Mathematics

Recommended Citation

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