Finite packing and covering by congruent convex domains
type: Item
instance of: scholarly article
title: Finite packing and covering by congruent convex domains
zbMATH Open document ID: 1046.52014
author: Károly jun. Böröczky
published in: Discrete \& Computational Geometry
publication date: 2004-03-07T00:00:00Z
review text: For a given convex domain \(K\), let \(H(K)\) be a circumscribed polygon with at most six sides whose area is minimal. According to the celebrated result by L.~Fejes Tóth, if a polygon \(P\) having at most six sides contains \(n\) non-overlapping congruent copies of \(K\), then for its area we have \(A(P)\geq n\cdot A(H(K))\). In this paper, the author provides a stronger version of this result: If a convex domain \(D\) contains \(n\) non-overlapping congruent copies of a convex domain \(K\), then \(A(D)\geq n\cdot A(H(K))\) provided that \(n\geq N\) where \(N\) depends only on \(K\). It is shown that the \(N\) does depend on \(K\) (even for centrally symmetric domains), and cannot be chosen to be an absolute constant. The constant \(A(H(K))\) is not optimal for the typical convex domain \(K\) (in the sense of the Baire category). On the other hand, if \(K\) is centrally symmetric, then one can choose \(H(K)\) to be centrally symmetric according to C.~H.~Dowker. Thus there exists a lattice tiling by translates of \(H(K)\), and hence taking large domains as \(D\) shows that the constant \(A(H(K))\) is optimal in this case. As a corollary, the author obtains the following. For a centrally symmetric convex domain \(K\) that is not a parallelogram, let \(D_n\) be a convex domain with minimal area that contains \(n\) non-overlapping congruent copies of \(K\). Then \(c_1\cdot\sqrt{n} < r(D_n) < R(D_n) < c_2 \cdot\sqrt{n},\) where the positive constants \(c_1\) and \(c_2\) depend on \(K\). In case of coverings some extra assumptions necessary: two convex domains \(C_1\) and \(C_2\) are called non-crossing if there exist complementary half-planes \(I^-\) and \(I^+\) such that \(I^-\cap C_1 \subset C_2\) and \(I^+\cap C_2 \subset C_1\). For a convex domain \(K\), let \(\widetilde H(K)\) be an inscribed hexagon with at most six sides whose area is maximal. If a convex domain \(D\) is covered by \(n\) non-crossing congruent copies of a convex domain \(K\), then \(A(D)\leq n\cd…
Mathematics Subject Classification ID: 52C20
Mathematics Subject Classification ID: 52C15
zbMATH DE Number: 2051388
zbMATH Keywords: convex domain
zbMATH Keywords: minimum circumscribed polygon
zbMATH Keywords: centrally symmetric convex domain
zbMATH Keywords: imitating a cell decomposition
zbMATH Keywords: packing congruent domains
zbMATH Keywords: covering by non-crossing congruent domains
zbMATH Keywords: lattice tiling
reviewed by: Serguey M. Pokas
MaRDI profile type: Publication
full work available at URL: s00454-003-0005-8
OpenAlex ID: W2906727341
cites work: Covering the Plane with Convex Sets
cites work: On minimum circumscribed polygons
cites work: Finite coverings by translates of centrally symmetric convex domains
cites work: Densest packings of typical convex sets are not lattice-like
cites work: Q4328182
cites work: Q5795745
cites work: Q4274034
DOI: 10.1007/S00454-003-0005-8
description: scientific article; zbMATH DE number 2051388