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
Hits: 3030

We use cookies to improve our website and your experience when using it. Cookies used for the essential operation of this site have already been set. To find out more about the cookies we use and how to delete them, see our privacy policy.

  I accept cookies from this site.
EU Cookie Directive Module Information