Positivity of Curvature and Convexity of Faces

M. I. Shtogrin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology, discrete geometry, and set theory, Tome 252 (2006), pp. 277-284

Résumé

Two-dimensional polyhedra homeomorphic to closed two-dimensional surfaces are considered in the three-dimensional Euclidean space. While studying the structure of an arbitrary face of a polyhedron, an interesting particular case is revealed when the magnitude of only one plane angle determines the sign of the curvature of the polyhedron at the vertex of this angle. Due to this observation, the following main theorem of the paper is obtained: If a two-dimensional polyhedron in the three-dimensional Euclidean space is isometric to the surface of a closed convex three-dimensional polyhedron, then all faces of the polyhedron are convex polygons.