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Journal for Geometry and Graphics 13 (2009), No. 1, 059--073
Copyright Heldermann Verlag 2009

On the Combinatorics of Inflexion Arches of Saddle Spheres

Gaiane Panina
Institute for Informatics and Automation, Russian Academy of Sciences, 14 linia 39, 199178 St. Petersburg, Russia


Each saddle sphere $\Gamma \subset S^3$ is known to generate a spanning arrangement of at least four non-crossing oriented great semicircles on $S^2$. Each semicircle arises as the projection of an inflexion arch of the surface $\Gamma$. In the paper we prove the converse: each spanning arrangement of non-crossing oriented great semicircles is generated by some smooth saddle sphere. In particular, this means the diversity of saddle spheres on $S^3$. Recall that each $C^2$-smooth saddle sphere leads directly to a counterexample to the following conjecture of A. D. Alexandrov: \par {\it Let $K \subset \mathbb{R}^3$ be a smooth convex body. If, for a constant $C$, at every point of $\partial K$, we have $R_1 \leq C \leq R_2$, then $K$ is a ball ($R_1$ and $R_2$ stand for the principal curvature radii of $\partial K$).} \par In the framework of the conjecture, the main result of the paper means that all counterexamples can be classified by non-crossing arrangements of oriented great semicircles.

Keywords: Alexandrov's conjecture, inflexion point, inflexion arch, saddle surface, hyperbolic virtual polytope.

MSC: 53C45; 53A10

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