View-obstruction problem for 3-dimensional spheres
type: Item
instance of: scholarly article
title: View-obstruction problem for 3-dimensional spheres
zbMATH Open document ID: 0588.10034
DOI: 10.1007/BF01559391
author: V. C. Dumir
author: R. J. Hans-Gill
published in: Monatshefte für Mathematik
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178279
review text: The view-obstruction problem for an n-dimensional closed convex body C (containing the origin in its interior) has been formulated by \textit{T. W. Cusick} [Aequationes Math. 9, 165-170 (1973; Zbl 0265.52003)]. The problem is to determine the constant K(C) defined to be the infimum of those \(\alpha >0\), such that for any ray L starting from the origin and lying in the positive quadrant, there are non-negative integers \(m_ 1,...,m_ n\) for which L intersects \(\alpha C+(m_ 1+,...,m_ n+)\). K(C) is denoted by \(\nu\) (n) when C is the n-dimensional sphere with centre 0 and diameter 1. Here it is shown that \(\nu (3)=\sqrt{3/7}\) thereby proving a conjecture of Cusick (loc. cit.). It is further shown that the result is isolated and a complete chain of isolated minima converging to 1/\(\sqrt{3}\) is obtained. The constant 1/\(\sqrt{3}\) is needed for uncountably many rays. In the proof, the authors use isolation results for the view-obstruction problem for circles and a variant (see lemmas 1 and 2). The authors expect that analogous isolation results hold for the cube with centre 0 in \({\mathbb{R}}^ 3\).
Mathematics Subject Classification ID: 11J99
Mathematics Subject Classification ID: 52C07
Mathematics Subject Classification ID: 52C17
zbMATH DE Number: 3943930
zbMATH Keywords: view-obstruction problem
zbMATH Keywords: closed convex body
zbMATH Keywords: n-dimensional sphere
zbMATH Keywords: isolation results
MaRDI profile type: Publication
cites work: Untere Schranken für zwei diophantische Approximations-Funktionen
cites work: View-obstruction problems
cites work: View-obstruction problems in \(n\)-dimensional geometry
cites work: View-Obstruction Problems. II
cites work: View-obstruction problems. III
cites work: Zur simultanen homogenen diophantischen Approximation. I, II, III
OpenAlex ID: W1997584434
description: scientific article; zbMATH DE number 3943930