Article
Licensed
Unlicensed
Requires Authentication
New counterexamples to A. D. Alexandrov’s hypothesis
-
Gaiane Panina
Published/Copyright:
July 27, 2005
Abstract
The paper presents a series of principally different C∞ -smooth counterexamples to the following hypothesis on a characterization of the sphere: Let K ⊂ ℝ3 be a smooth convex body. If at every point of ∂K, we have R1≤ C≤ R2 for a constant C, then K is a ball. (R1 and R2 stand for the principal curvature radii of ∂K.)
The hypothesis was proved by A. D. Alexandrov and H. F. Münzner for analytic bodies. For the case of general smoothness it has been an open problem for years. Recently, Y. Martinez-Maure has presented a C2-smooth counterexample to the hypothesis.
:
Published Online: 2005-07-27
Published in Print: 2005-04-20
© de Gruyter
You are currently not able to access this content.
You are currently not able to access this content.
Articles in the same Issue
- Localization of automorphisms of some unbounded Levi degenerate algebraic hypersurfaces in ℂn
- Virtual and non-virtual algebraic Betti numbers
- The classification of flag-transitive Steiner 3-designs
- On surfaces with two apparent double points
- Halphen conditions and postulation of nodes
- Topological affine planes with affine connections
- The embedding of (0, 2)-geometries and semipartial geometries in AG(n, q)
- The geometry of isoparametric hypersurfaces with four distinct principal curvatures in spheres
- New counterexamples to A. D. Alexandrov’s hypothesis
- A maximum principle for parabolic equations on manifolds with cone singularities
- Packing a planar convex body with three homothetical copies and inscribing relatively equilateral triangles
Articles in the same Issue
- Localization of automorphisms of some unbounded Levi degenerate algebraic hypersurfaces in ℂn
- Virtual and non-virtual algebraic Betti numbers
- The classification of flag-transitive Steiner 3-designs
- On surfaces with two apparent double points
- Halphen conditions and postulation of nodes
- Topological affine planes with affine connections
- The embedding of (0, 2)-geometries and semipartial geometries in AG(n, q)
- The geometry of isoparametric hypersurfaces with four distinct principal curvatures in spheres
- New counterexamples to A. D. Alexandrov’s hypothesis
- A maximum principle for parabolic equations on manifolds with cone singularities
- Packing a planar convex body with three homothetical copies and inscribing relatively equilateral triangles
