A combinatorial result about points and balls in euclidean space

Abstract

For each n≥1 there is cn>0 such that for any finite sex X ⊆ ℝ″ there is A ⊆X, |A|≤1/2(n+3), having the following property: if B ⊇A is an n-ball, then |B ∩X|≥cn|X|. This generalizes a theorem of Neumann-Lara and Urrutia which states that c2≥1/60. © 1989 Springer-Verlag New York Inc.