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A CHARACTERIZATION OF ELLIPTIC HYPERBOLOIDS

Kim, Dong-Soo;Son, Booseon

  • Received : 2012.12.18
  • Accepted : 2013.01.21
  • Published : 2013.03.25

Abstract

Consider a non-degenerate open convex cone C with vertex the origin in the $n$2-dimensional Euclidean space $E^n$. We study volume properties of strictly convex hypersurfaces in the cone C. As a result, for example, if the volume of the region of an elliptic cone C cut off by the tangent hyperplane P of M at $p$ is independent of the point $p{\in}M$, then it is shown that the hypersurface M is part of an elliptic hyperboloid.

Keywords

strictly convex hypersurface;elliptic hyperboloid;homogeneous extension;n-dimensional volume;elliptic cone;support function

References

  1. Kim, D.-S. and Kim, Y. H., Some characterizations of spheres and elliptic paraboloids, Linear Algebra Appl. 437 (2012), no. 1, 113-120. https://doi.org/10.1016/j.laa.2012.02.013
  2. Kim, D.-S. and Kim, Y. H., Some characterizations of spheres and elliptic paraboloids II, Linear Algebra Appl., 438 (2013), no. 3, 1356-1364. https://doi.org/10.1016/j.laa.2012.08.024
  3. Kim, D.-S., Ellipsoids and elliptic hyperboloids in the Euclidean space $\mathbb{E}^{n+1}$, Submitted.
  4. O'Neill, B., Elementary differential geometry, Revised second edition, Elsevier/Academic Press, Amsterdam, 2006.
  5. Kim, D.-S. and Kim, Y. H., A characterization of ellipses, Amer. Math. Monthly 114 (2007), no. 1, 66-70.

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