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Difference between revisions of "Limit cone"

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''of a convex surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058840/l0588401.png" />''
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''of a convex surface $S$''
  
The surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058840/l0588402.png" /> of the cone formed by the straight half-lines emanating from some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058840/l0588403.png" /> and belonging to the convex body bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058840/l0588404.png" />. The limit cone is defined uniquely, up to a parallel displacement depending on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058840/l0588405.png" />. The notion of a limit cone is also defined for certain classes of non-convex surfaces, e.g. for so-called spherical one-sheet saddle surfaces.
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The surface $P(S)$ of the cone formed by the straight half-lines emanating from some point $O\in S$ and belonging to the convex body bounded by $S$. The limit cone is defined uniquely, up to a parallel displacement depending on the choice of $O$. The notion of a limit cone is also defined for certain classes of non-convex surfaces, e.g. for so-called spherical one-sheet saddle surfaces.
  
 
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Latest revision as of 05:57, 18 October 2014

of a convex surface $S$

The surface $P(S)$ of the cone formed by the straight half-lines emanating from some point $O\in S$ and belonging to the convex body bounded by $S$. The limit cone is defined uniquely, up to a parallel displacement depending on the choice of $O$. The notion of a limit cone is also defined for certain classes of non-convex surfaces, e.g. for so-called spherical one-sheet saddle surfaces.

References

[1] A.V. Pogorelov, "Extrinsic geometry of convex surfaces" , Amer. Math. Soc. (1972) (Translated from Russian)


Comments

References

[a1] R. Schneider, "Boundary structure and curvature of convex bodies" J. Tölke (ed.) J.M. Wills (ed.) , Contributions to geometry , Birkhäuser (1979) pp. 13–59
[a2] H. Busemann, "Convex surfaces" , Interscience (1958) pp. Chapt. III, Sect. 15
How to Cite This Entry:
Limit cone. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Limit_cone&oldid=17176
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article