On a covering problem in the plane
type: Item
instance of: scholarly article
title: On a covering problem in the plane
zbMATH Open document ID: 0976.52002
published in: Beiträge zur Algebra und Geometrie
publication date: 2000-09-13T00:00:00Z
full work available at URL: https://eudml.org/doc/121450
full work available at URL: 12.html
review text: Referring to a problem (p. 392: Problem 28) posed in the book ``Excursions into combinatorial Geometry'', Berlin/Heidelberg/New York (1997; Zbl 0877.52001), by \textit{V. Boltyanski, H. Martini} and \textit{P. S. Soltan}, the author proves the Theorem: There are (sets which are) minimal covering sets for all planar sets of constant width \(d\), except for closed Reuleaux triangles. To prove this, denote by \(D\) a Reuleaux triangle of width \(d\) and consider either a regular hexagon \(D_1\) whose opposite sides have the distance \(d\), or a convex domain \(D_2\) the boundary of which consists of two arcs of the boundary of \(D\) and a semicircle (H. G. Eggleston, 1963). \(D\) can be embedded in \(D_i\) in \(3-i\) positions (\(i=1,2\)). If one denotes the set of the corners of \(D_i\) by \(C_i\), then \(D_i\setminus C_i\) cannot cover a closed Reuleaux triangle \(D\), but -- on the other hand -- one can cover each planar set of constant width \(d\) different from the Reuleaux triangles with the sets \(D_i\setminus C_i\) (only the Reuleaux triangles have corners with a projection cone of the measure \(2\pi/3\)). By Zorn's lemma it follows that \(D_i\setminus C_i\) must contain a minimal set with the same covering properties. In consequence of the theorem above, the questions of ``Problem 28'' (among them: ``Is any minimal, convex covering set for the class of all figures of constant width 1 also a minimal, convex covering set for the class of all Reuleaux polygons of width 1, and vice versa?'') must be reformulated as covering problems for open sets (of constant width).
Mathematics Subject Classification ID: 52A10
zbMATH DE Number: 1505615
zbMATH Keywords: minimal convex covering set
zbMATH Keywords: planar set of constant width
zbMATH Keywords: Reuleaux polygon
zbMATH Keywords: Reuleaux triangle
author: Bernulf Weissbach
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1505615