Steklov Institute of Mathematics at St.Petersburg

PREPRINT 04/2003

NEW COUNTEREXAMPLES TO AN OLD UNIQUENESS HYPOTHESIS FOR CONVEX BODIES

This preprint was accepted May 12, 2003

ABSTRACT:
The paper presents a series of principally different $C^\infty$ -smooth counterexamples
to the hypothesis on characterization of the sphere:

If for a smooth convex body $K \subset \Bbb R^3$ and a constant $C$ ,
 in each point of $ \partial K$ we have $R_1 \leq C \leq R_2$,
then $K$ is a ball. ($R_1$ and $R_2$ are the principal curvature radii of $\partial K$).


The hypothesis was proved by A.D.Alexandrov and H.F.M\"unzner for analitic bodies.
 For the general smooth case  it remained an open problem for years.
Recently, Y.~Martinez--Maure presented a $C^2$-smooth counterexample to the hypothesis.