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Search: MSC category 52B99 ( None of the above, but in this section )

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1. CJM Online first

Izmestiev, Ivan
 Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the Hilbert-Einstein Functional The paper is centered around a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert-Einstein functional on the space of "warped polyhedra" with a fixed metric on the boundary. The situation is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra. This latter kind of rigidity is related to the Minkowski theorem on the existence and uniqueness of a polyhedron with prescribed face normals and face areas. In the spherical and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert-Einstein functional and the volume, as well as between both kinds of rigidity. We review some of the related work and discuss directions for future research. Keywords:convex polyhedron, rigidity, Hilbert-Einstein functional, Minkowski theoremCategories:52B99, 53C24

2. CJM 2010 (vol 62 pp. 975)

Bjorndahl, Christina; Karshon, Yael
 Revisiting Tietze-Nakajima: Local and Global Convexity for Maps A theorem of Tietze and Nakajima, from 1928, asserts that if a subset $X$ of $\mathbb{R}^n$ is closed, connected, and locally convex, then it is convex. We give an analogous local to global convexity" theorem when the inclusion map of $X$ to $\mathbb{R}^n$ is replaced by a map from a topological space $X$ to $\mathbb{R}^n$ that satisfies certain local properties. Our motivation comes from the Condevaux--Dazord--Molino proof of the Atiyah--Guillemin--Sternberg convexity theorem in symplectic geometry. Categories:53D20, 52B99

3. CJM 2004 (vol 56 pp. 472)