Finite coverings by translates of centrally symmetric convex domains
type: Item
instance of: scholarly article
title: Finite coverings by translates of centrally symmetric convex domains
zbMATH Open document ID: 0632.52011
DOI: 10.1007/BF02187889
author: B. George
published in: Discrete \& Computational Geometry
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/131028
review text: A convex domain is a compact convex subset of the Euclidean plane with non-empty interior. Let C be a fixed convex domain which has a centre, and let q and h be the maximum area of a quadrangle and the maximum area of the hexagon, both contained in C. The area of a set K is denoted by a(K), and the area of C by a. The following theorem is proved: If a convex domain K is covered by n translates of C then \[ a(K) \leq (n-1)h + a - \frac{h-q}{2(a+q)}A, \] where \[ A = \{(a+h)^ 2+ (a+q)[4(n-1)h+3a- q]\}^{1/2}- a - h. \] This improves upon the inequality \(a(K)\leq (n- 1)h+a\) obtained by \textit{R. P. Bambah} and \textit{C. A. Rogers} [J. Lond. Math. Soc. 27, 304-314 (1952; Zbl 0046.380)]. Let \(\theta_ n(C)= \inf \{na/a(K)\), where K can be covered by n translates of \(C\}\), and let \(\theta\) (C) be the least density of covering of the plane by translates of C. The above theorem implies that \(\theta_ n(C)>\theta (C)\) for any central convex domain C other than a parallelogram, and any integer \(n\geq 26\).
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 05B40
zbMATH DE Number: 4028294
zbMATH Keywords: translates of a domain
zbMATH Keywords: central domain
zbMATH Keywords: convex domain
reviewed by: August Florian
MaRDI profile type: Publication
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OpenAlex ID: W2110431853
description: scientific article; zbMATH DE number 4028294