Generalized breadths, circular Cantor sets, and the least area UCC
type: Item
instance of: scholarly article
title: Generalized breadths, circular Cantor sets, and the least area UCC
zbMATH Open document ID: 0817.52005
DOI: 10.1007/BF02574391
published in: Discrete \& Computational Geometry
publication date: 1995-08-01T00:00:00Z
full work available at URL: https://eudml.org/doc/131341
review text: A universal convex cover (UCC for short) is a convex set in \(\mathbb{R}^ 2\) which, after shifting and rotating, can cover any point set of unit diameter. In this short and very readable paper, the author takes on an old question by Lebesgue, of determining the least area UCC. The smallest known UCC, of area slightly less than 0.8441 is provided by \textit{H. C. Hansen} [Geom. Dedicata 42, No. 2, 205-213 (1992; Zbl 0752.52011)]. This paper provides a simple construction (circular Cantor sets) improving the best known lower bound on the area of a UCC (with an area of slightly more than 0.8257). The paper's best construction gives a lower bound of \({3 \over 4} [\tan {\pi \over 6} + \sum_{i \geq 0} 2^ i \tan {\pi \over 18.3^ i}]\), which is slightly more than 0.8271.
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 698614
zbMATH Keywords: convex cover
zbMATH Keywords: convex set
author: György Elekes
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 698614