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arXiv · 1502.07869

Subtended Angles

Paul Balister·Béla Bollobás·Zoltán Füredi·Imre Leader·Mark Walters
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Abstract

We consider the following question. Suppose that $d\ge2$ and $n$ are fixed, and that $\theta_1,\theta_2,\dots,\theta_n$ are $n$ specified angles. How many points do we need to place in $\mathbb{R}^d$ to realise all of these angles? A simple degrees of freedom argument shows that $m$ points in $\mathbb{R}^2$ cannot realise more than $2m-4$ general angles. We give a construction to show that this bound is sharp when $m\ge 5$. In $d$ dimensions the degrees of freedom argument gives an upper bound of $dm-\binom{d+1}{2}-1$ 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 $2m-3$ of angles that cannot be realised by $m$ points in any dimension.

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Paul Balister, Béla Bollobás, Zoltán Füredi, Imre Leader, Mark Walters. 2015-02-27. Subtended Angles. https://arxiv.org/abs/1502.07869

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