Subtended Angles

Balister, Paul; Bollobás, Béla; F\, Zoltán
We consider the following question. Suppose that {\$}d\backslashge2{\$} and {\$}n{\$} are fixed, and that {\$}\backslashtheta{\_}1,\backslashtheta{\_}2,\backslashdots,\backslashtheta{\_}n{\$} are {\$}n{\$} specified angles. How many points do we need to place in {\$}\backslashmathbb{\{}R{\}}{\^{}}d{\$} to realise all of these angles? A simple degrees of freedom argument shows that {\$}m{\$} points in {\$}\backslashmathbb{\{}R{\}}{\^{}}2{\$} cannot realise more than {\$}2m-4{\$} general angles. We give a construction to show that this bound is sharp when {\$}m\backslashge 5{\$}. In {\$}d{\$} dimensions the degrees of freedom argument gives an upper bound of {\$}dm-\backslashbinom{\{}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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