Abstract
To points p and q of a finite set S in d-dimensional Euclidean space E d are extreme if { p, q} = S ∩ h, for some open halfspace h. Let e 2 ( d) ( n) be the maximum number of extreme pairs realized by any n points in E d . We give geometric proofs of e 2 (2)(n) = ⌊ 3n 2 ⌋ , if n⩾4, and e 2 (3)( n) = 3 n−6, if n⩾6. These results settle the question since all other cases are trivial.
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