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Sbornik: Mathematics, 2013, Volume 204, Issue 8, Pages 1122–1130
DOI: https://doi.org/10.1070/SM2013v204n08ABEH004332 (Mi sm8169) 		 
This article is cited in 3 scientific papers (total in 3 papers)

On the structure of self-affine convex bodies

A. S. Voynov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
English version PDF (457 kB) Citations (3)
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DOI: https://doi.org/10.1070/SM2013v204n08ABEH004332
Abstract: We study the structure of convex bodies in $\mathbb R^d$ that can be represented as a union of their affine images with no common interior points. Such bodies are called self-affine. Vallet's conjecture on the structure of self-affine bodies was proved for $d = 2$ by Richter in 2011. In the present paper we disprove the conjecture for all $d \geqslant 3$ and derive a detailed description of self-affine bodies in $\mathbb R^3$. Also we consider the relation between properties of self-affine bodies and functional equations with a contraction of an argument.
Bibliography: 10 titles.
Keywords: partition, self-affine set, convex polyhedron.
Funding agency Grant number
Russian Foundation for Basic Research 11-01-00329
12-01-31281-мол_а
Dynasty Foundation
Ministry of Education and Science of the Russian Federation 8215
Received: 30.08.2012
Russian version:
Matematicheskii Sbornik, 2013, Volume 204, Number 8, Pages 41–50
DOI: https://doi.org/10.4213/sm8169
Bibliographic databases:
Document Type: Article
UDC: 514.172.45+514.174.5
MSC: 52A99
Language: English
Original paper language: Russian
Citation: A. S. Voynov, “On the structure of self-affine convex bodies”, Mat. Sb., 204:8 (2013), 41–50; Sb. Math., 204:8 (2013), 1122–1130
Citation in format AMSBIB
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  • https://www.mathnet.ru/eng/sm/v204/i8/p41
    • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Abstract page:400
    Russian version PDF:182
    English version PDF:7
    References:48
    First page:24
     
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