The paper presents a series of principally di¤erent C y -smooth counterexamples to the following hypothesis on a characterization of the sphere: Let K H R 3 be a smooth convex body. If at every point of qK, we have R 1 c C c R 2 for a constant C, then K is a ball. (R 1 and R 2 stand for the principal curvature radii of qK.)The hypothesis was proved by A. D. Alexandrov and H. F. Mü nzner for analytic bodies. For the case of general smoothness it has been an open problem for years. Recently, Y. Martinez-Maure has presented a C 2 -smooth counterexample to the hypothesis.

Abstract. We describe and study an explicit structure of a regular cell complex K(L) on the moduli space M (L) of a planar polygonal linkage L. The combinatorics is very much related (but not equal) to the combinatorics of the permutohedron. In particular, the cells of maximal dimension are labeled by elements of the symmetric group. For example, if the moduli space M is a sphere, the complex K is dual to the boundary complex of the permutohedron.The dual complex K * is patched of Cartesian products of permutohedra. It can be explicitly realized in the Euclidean space via a surgery on the permutohedron.

It is known that cyclic configurations of a planar polygonal linkage are critical points of the signed area function. In the paper, we announce an explicit formula of the Morse index for the signed area of a cyclic configuration.It depends not only on the combinatorics of a cyclic configuration, but also includes some metric characterization.

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Equilibria of polygonal linkage with respect to Coulomb potential of point charges placed at the vertices of linkage are considered. It is proved that any convex configuration of a quadrilateral linkage is the point of global minimum of Coulomb potential for appropriate values of charges of vertices. Similar problems are treated for the equilateral pentagonal linkage. Some corollaries and applications in the spirit of control theory are also presented.