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Izv. RAN. Ser. Mat., 2005, Volume 69, Issue 4, Pages 205–224 (Mi izv653)  

This article is cited in 2 scientific papers (total in 2 papers)

On the structure of faces of three-dimensional polytopes

M. I. Shtogrin

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: In Euclidean three-space, we consider two-dimensional polyhedra that are homeomorphic to closed surfaces. The structure of an arbitrary face of such a polyhedron is studied in detail. In particular, we prove the following main theorem. If a two-dimensional polyhedron lies in Euclidean three-space and is isometric to the surface of a convex three-dimensional polytope, then all the faces of the polyhedron are convex polygons.

DOI: https://doi.org/10.4213/im653

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English version:
Izvestiya: Mathematics, 2005, 69:4, 205–224

Bibliographic databases:

UDC: 514.113.5+514.172.45
MSC: 52B10, 52C25, 53C45
Received: 19.01.2005

Citation: M. I. Shtogrin, “On the structure of faces of three-dimensional polytopes”, Izv. RAN. Ser. Mat., 69:4 (2005), 205–224; Izv. Math., 69:4 (2005), 205–224

Citation in format AMSBIB
\Bibitem{Sht05}
\by M.~I.~Shtogrin
\paper On~the structure of~faces of~three-dimensional polytopes
\jour Izv. RAN. Ser. Mat.
\yr 2005
\vol 69
\issue 4
\pages 205--224
\mathnet{http://mi.mathnet.ru/izv653}
\crossref{https://doi.org/10.4213/im653}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2170708}
\zmath{https://zbmath.org/?q=an:1085.52006}
\elib{http://elibrary.ru/item.asp?id=9195229}
\transl
\jour Izv. Math.
\yr 2005
\vol 69
\issue 4
\pages 205--224
\crossref{https://doi.org/10.1070/IM2005v069n04ABEH001667}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33645516361}


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  • https://doi.org/10.4213/im653
  • http://mi.mathnet.ru/eng/izv/v69/i4/p205

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. M. I. Shtogrin, “Positivity of Curvature and Convexity of Faces”, Proc. Steklov Inst. Math., 252 (2006), 264–271  mathnet  crossref  mathscinet  elib
    2. M. I. Shtogrin, “Piecewise Smooth Developable Surfaces”, Proc. Steklov Inst. Math., 263 (2008), 214–235  mathnet  crossref  mathscinet  zmath  isi  elib  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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