Cutting corners with spheres in \(d\)-dimensions
Let \(\mathbb{N}_0\) denote the set of non-negative integers. A sphere corner cut \(A \subset \mathbb{N}_0^d\) is a set of points that includes the origin and can be separated from \(\mathbb{N}_0^d \backslash A\) by a sphere. In other words, \(A\) is a sphere corner cut if there are real numbers \(a_1,a_2,\dots,a_d\) and \(r\) such that \[ \sum_{i=1}^d (x_i - a_i)^2 < r^2 \] for all \( x \in A\) and \[ \sum_{i=1}^d (y_i - a_i)^2 > r^2 \] for all \(y \in \mathbb{N}_0^d \backslash A\). It is shown in this paper that the number of sphere corner cuts having \(n\) points is \[ O(n^{d+1}(\log n)^{d-1}). \] The method of proof can also be applied to cutting corners by hyperplanes, and in this case it is found that the number of hyperplane corner cuts having \(n\) points is \[ O(n^{d-1/d}(\log n)^{d-1}). \]
scientific article; zbMATH DE number 2107253