### A characterization of compact convex polyhedra in hyperbolic 3-space.

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We present a constructive proof of Alexandrov’s theorem on the existence of a convex polytope with a given metric on the boundary. The polytope is obtained by deforming certain generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a connection with weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of generalized convex...

The paper gives an illustrated introduction to the theory of hyperbolic virtual polytopes and related counterexamples to A.D. Alexandrov’s conjecture.

Let ${e}_{ij}$ be the number of edges in a convex 3-polytope joining the vertices of degree i with the vertices of degree j. We prove that for every convex 3-polytope there is $20{e}_{3,3}+25{e}_{3,4}+16{e}_{3,5}+10{e}_{3,6}+6[2/3]{e}_{3,7}+5{e}_{3,8}+2[1/2]{e}_{3,9}+2{e}_{3,10}+16[2/3]{e}_{4,4}+11{e}_{4,5}+5{e}_{4,6}+1[2/3]{e}_{4,7}+5[1/3]{e}_{5,5}+2{e}_{5,6}\ge 120$; moreover, each coefficient is the best possible. This result brings a final answer to the conjecture raised by B. Grünbaum in 1973.

In 1970, E.M.Andreev published a classification of all three-dimensional compact hyperbolic polyhedra (other than tetrahedra) having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, $C$, Andreev’s Theorem provides five classes of linear inequalities, depending on $C$, for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing $C$ with the assigned dihedral angles. Andreev’s Theorem also shows that the resulting...

Given a combinatorial description C of a polyhedron having E edges, the space of dihedral angles of all compact hyperbolic polyhedra that realize C is generally not a convex subset of RE. If C has five or more faces, Andreev's Theorem states that the corresponding space of dihedral angles AC obtained by restricting to non-obtuse angles is a convex polytope. In this paper we explain why Andreev did not consider tetrahedra, the only polyhedra having fewer than five faces, by demonstrating that the...

We investigate countably convex ${G}_{\delta}$ subsets of Banach spaces. A subset of a linear space is countably convex if it can be represented as a countable union of convex sets. A known sufficient condition for countable convexity of an arbitrary subset of a separable normed space is that it does not contain a semi-clique [9]. A semi-clique in a set S is a subset P ⊆ S so that for every x ∈ P and open neighborhood u of x there exists a finite set X ⊆ P ∩ u such that conv(X) ⊈ S. For closed sets this condition...