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

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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 relate the total curvature and the isoperimetric deficit of a curve $\gamma $ in a two-dimensional space of constant curvature with the area enclosed by the evolute of $\gamma $. We provide also a Gauss-Bonnet theorem for a special class of evolutes.

A hyperideal polyhedron is a non-compact polyhedron in the hyperbolic $3$-space ${\mathbb{H}}^{3}$ which, in the projective model for ${\mathbb{H}}^{3}\subset {\mathrm{\mathbb{R}\mathbb{P}}}^{3}$, is just the intersection of ${\mathbb{H}}^{3}$ with a projective polyhedron whose vertices are all outside ${\mathbb{H}}^{3}$ and whose edges all meet ${\mathbb{H}}^{3}$. We classify hyperideal polyhedra, up to isometries of ${\mathbb{H}}^{3}$, in terms of their combinatorial type and of their dihedral angles.

We study various aspects of nonexpansive retracts and retractions in certain Banach and metric spaces, with special emphasis on the compact nonexpansive envelope property.