We show that the following two (quite different!) problems have one and the same combinatorial background.
1. Carpenter's rule problem.
Can any carpenter's ruler (a closed planar non-crossing closed broken line) be straightened in the plane avoiding self-crossing?
2. A.D.Alexandrov's problem.
Given a smooth convex 3D-body K and a constant C which separates the principal curvature radii of K (i.e., R1 \leq C \leq R2 everywhere), is K necessarily a ball?
The talk is based on papers by R.Connelly, Y.Martinez-Maure, D.Orden, G.Rota, B.Servatius, H.Servatius, I.Streinu, W.Whiteley, the speaker, and others.
For details, see http://www.arxiv.org/abs/math.MG/0607171.
Mauricio Peixoto intoduced the notion of Focal Decomposition as a Geometric Object related to investigation of ODE of the second order. We are going to intoduce a notion of Focal Equivalence of equations and to solve some classification problems related to this equivalence relation.
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