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.
Research areas:
Year:
2015
Type of Publication:
Article
Number:
317532
Pages:
1-15
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