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Tr. Mat. Inst. Steklova, 2006, Volume 252, Pages 277–284 (Mi tm77)  

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

Positivity of Curvature and Convexity of Faces

M. I. Shtogrin

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Two-dimensional polyhedra homeomorphic to closed two-dimensional surfaces are considered in the three-dimensional Euclidean space. While studying the structure of an arbitrary face of a polyhedron, an interesting particular case is revealed when the magnitude of only one plane angle determines the sign of the curvature of the polyhedron at the vertex of this angle. Due to this observation, the following main theorem of the paper is obtained: If a two-dimensional polyhedron in the three-dimensional Euclidean space is isometric to the surface of a closed convex three-dimensional polyhedron, then all faces of the polyhedron are convex polygons.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2006, 252, 264–271

Bibliographic databases:

UDC: 514.113.5+514.172.45
Received in May 2005

Citation: M. I. Shtogrin, “Positivity of Curvature and Convexity of Faces”, Geometric topology, discrete geometry, and set theory, Collected papers, Tr. Mat. Inst. Steklova, 252, Nauka, MAIK Nauka/Inteperiodika, M., 2006, 277–284; Proc. Steklov Inst. Math., 252 (2006), 264–271

Citation in format AMSBIB
\Bibitem{Sht06}
\by M.~I.~Shtogrin
\paper Positivity of Curvature and Convexity of Faces
\inbook Geometric topology, discrete geometry, and set theory
\bookinfo Collected papers
\serial Tr. Mat. Inst. Steklova
\yr 2006
\vol 252
\pages 277--284
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm77}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2255984}
\elib{http://elibrary.ru/item.asp?id=13506342}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2006
\vol 252
\pages 264--271
\crossref{https://doi.org/10.1134/S0081543806010226}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33746073336}


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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, “Piecewise Smooth Developable Surfaces”, Proc. Steklov Inst. Math., 263 (2008), 214–235  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    2. M. I. Shtogrin, “Bending of a piecewise developable surface”, Proc. Steklov Inst. Math., 275 (2011), 133–154  mathnet  crossref  mathscinet  isi  elib  elib
  •    . . .  Proceedings of the Steklov Institute of Mathematics
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