Cutting corners with spheres in \(d\)-dimensions
type: Item
instance of: scholarly article
title: Cutting corners with spheres in \(d\)-dimensions
zbMATH Open document ID: 1056.52006
DOI: 10.1016/S0196-8858(03)00094-0
published in: Advances in Applied Mathematics
publication date: 2004-10-12T00:00:00Z
review text: 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 \[ O(n^{d-1/d}(\log n)^{d-1}). \]
reviewed by: Jeremy Lovejoy
Mathematics Subject Classification ID: 52B05
Mathematics Subject Classification ID: 52B55
zbMATH DE Number: 2107253
zbMATH Keywords: sphere corner cuts
zbMATH Keywords: hyperplanes
zbMATH Keywords: corner cuts
Wikidata QID: Q57944037
author: Joviša Žunić
MaRDI profile type: Publication
cites work: On the number of digital straight line segments
cites work: The number of plane corner cuts
cites work: On a Principle of Lipschitz
cites work: Q3994503
cites work: Cutting corners
cites work: On the number of corner cuts
cites work: A General Coding Scheme for Families of Digital Curve Segments
description: scientific article; zbMATH DE number 2107253