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Mat. Sb., 2010, Volume 201, Number 8, Pages 127–160 (Mi msb6369)  

This article is cited in 4 scientific papers (total in 4 papers)

Covering planar sets

V. P. Filimonov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Problems connected with the classical Borsuk problem on partitioning a set in Euclidean space into subsets of smaller diameter, and also connected with the Nelson-Hadwiger problem on the chromatic number of Euclidean space, are studied. New bounds are obtained for the quantities $d_n=\sup d_n(\Phi)$ and $d'_n=\sup d'_n(\Phi)$, where the suprema are taken over all sets of unit diameter on a plane, and where the quantities $d_n(\Phi)$ and $d'_n(\Phi)$ are defined for a given bounded set $\Phi\subset\mathbb{R}^2$ as follows:
\begin{align*} d_n(\Phi)&=\inf\{x\in\mathbb{R}^+:\Phi\subseteq \Phi_1\cup…\cup\Phi_n, \forall  i \operatorname{diam}\Phi_i\le x\},
d'_n(\Phi)&=\inf\{x\in\mathbb{R}^+:\Phi\subseteq \Phi_1\cup…\cup\Phi_n, \forall  i \forall  X,Y\in\Phi_i  XY\ne x\}. \end{align*}
Here the $\Phi_i\subset\mathbb R^2$ are subsets, $\operatorname{diam}\Phi_i$ is the diameter of $\Phi_i$, $XY$ is the distance between the points $X$ and $Y$, and $n\in \mathbb N$. The bounds obtained for $d_n$ are better than any known before; this paper is the first to consider the values $d'_n$.
Bibliography: 19 titles.

Keywords: chromatic number, Borsuk problem, diameter of a set, coverings of planar sets, universal covering sets and systems.

DOI: https://doi.org/10.4213/sm6369

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English version:
Sbornik: Mathematics, 2010, 201:8, 1217–1248

Bibliographic databases:

Document Type: Article
UDC: 514.174
MSC: 52C15
Received: 27.05.2008 and 24.08.2009

Citation: V. P. Filimonov, “Covering planar sets”, Mat. Sb., 201:8 (2010), 127–160; Sb. Math., 201:8 (2010), 1217–1248

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Bulankina V.V., “O razbienii ploskikh mnozhestv na pyat chastei bez rasstoyaniya: $\sqrt{2-\sqrt{3}}$”, Tr. Moskovskogo fiziko-tekhnicheskogo instituta, 4:1-13 (2012), 56–72  elib
    2. Voronetskii E.Yu., “O razbienii ploskikh mnozhestv na chetyre, pyat i shest chastei bez dostatochno malenkikh rasstoyanii”, Tr. Moskovskogo fiziko-tekhnicheskogo instituta, 4:1-13 (2012), 73–76  elib
    3. Belov D., Aleksandrov N., “O razbienii ploskikh mnozhestv na shest chastei malogo diametra”, Tr. Moskovskogo fiziko-tekhnicheskogo instituta, 4:1-13 (2012), 77–80  elib
    4. V. P. Filimonov, “Covering sets in $\mathbb{R}^m$”, Sb. Math., 205:8 (2014), 1160–1200  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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