Gaiane Panina

On hyperbolic virtual polytopes and hyperbolic fans

AbstractHyperbolic virtual polytopes arose originally as polytopal versions of counterexamples to the following A.D.Alexandrov’s uniqueness conjecture: Let K ⊂ ℝ3be a smooth convex body. If for a constant C, at every point of ∂K, we have R1 ≤ C ≤ R2then K is a ball. (R1and R2stand for the principal curvature radii of ∂K.) This paper gives a new (in comparison with the previous construction by Y.Martinez-Maure and by G.Panina) series of counterexamples to the conjecture. In particular, a hyperbolic virtual polytope (and therefore, a hyperbolic hérisson) with odd an number of horns is constructed.Moreover, various properties of hyperbolic virtual polytopes and their fans are discussed.

Morse index of a cyclic polygon

It is known that cyclic configurations of a planar polygonal linkage are critical points of the signed area function. In the paper we give an explicit formula of the Morse index for the signed area of a cyclic configuration. We show that it depends not only on the combinatorics of a cyclic configuration, but also on its metric properties.

Moduli Space of a Planar Polygonal Linkage: A Combinatorial Description

We describe and study an explicit structure of a regular cell complex

An illustrated theory of hyperbolic virtual polytopes

\mathcal {K}(L)

Polygons with prescribed edge slopes: configuration space and extremal points of perimeter

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

On the area of constrained polygonal linkages

\mathcal {K}