Cutting Corners by Circles and Spheres

A circle corner cut A ⊂ No2 is a planar set of points with nonnegative integer coordinates which includes the origin and which can be separated from No2 \ A by a circle. In this paper we show that there are O(n3 · log n) different circle corner cuts consisting of n points. phere corner cut is defined as a set A ⊂ No3 of points with nonnegative integer coordinates which includes the origin and which can be separated from No3 \ A by a sphere, then there are O(n4 · (log n)2) different sphere corner cuts consisting of n points.