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The equal tangents property

  • J. Jerónimo-Castro EMAIL logo , G. Ruiz-Hernández and S. Tabachnikov
Published/Copyright: July 8, 2014
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Advances in Geometry
From the journal Advances in Geometry Volume 14 Issue 3

Abstract

Let M be a C2-smooth strictly convex closed surface in ℝ3 and denote by H the set of points x in the exterior of M such that all the tangent segments from x to M have equal lengths. In this note we prove that if H is either a closed surface containing M or a plane, then M is a Euclidean sphere. Moreover, we shall see that the situation in the Euclidean plane is very different.

Published Online: 2014-7-8
Published in Print: 2014-7-1

© 2014 by Walter de Gruyter Berlin/Boston

Articles in the same Issue

  1. Masthead
  2. On axiomatic definitions of non-discrete affine buildings
  3. On Clifford analysis for holomorphic mappings
  4. Universal points of convex bodies and bisectors in Minkowski spaces
  5. The equal tangents property
  6. Some theorems of harmonic maps for Finsler manifolds
  7. The regular Grünbaum polyhedron of genus 5
  8. On the existence of nilsolitons on 2-step nilpotent Lie groups
  9. Rotational linear Weingarten hypersurfaces in the Euclidean sphere Sn+1
  10. Spacelike hypersurfaces in anti-de Sitter space
  11. The group of strong symplectic homeomorphisms in the L∞-metric
  12. The total Betti number of the intersection of three real quadrics
  13. On the algebraic models of symmetric smooth manifolds
  14. Non-existence of tight neighborly triangulated manifolds with β1 = 2
https://doi.org/10.1515/advgeom-2013-0011

Articles in the same Issue

  1. Masthead
  2. On axiomatic definitions of non-discrete affine buildings
  3. On Clifford analysis for holomorphic mappings
  4. Universal points of convex bodies and bisectors in Minkowski spaces
  5. The equal tangents property
  6. Some theorems of harmonic maps for Finsler manifolds
  7. The regular Grünbaum polyhedron of genus 5
  8. On the existence of nilsolitons on 2-step nilpotent Lie groups
  9. Rotational linear Weingarten hypersurfaces in the Euclidean sphere Sn+1
  10. Spacelike hypersurfaces in anti-de Sitter space
  11. The group of strong symplectic homeomorphisms in the L∞-metric
  12. The total Betti number of the intersection of three real quadrics
  13. On the algebraic models of symmetric smooth manifolds
  14. Non-existence of tight neighborly triangulated manifolds with β1 = 2
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