Intrinsic Geometry of Convex Surfaces
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A.D. Alexandrov is considered by many to be the father of intrinsic geometry. A two-volume set, A.D. Alexandrov Selected Works contains some of the most important papers by this renowned geometer. Volume 1 covers convex polyhedrons and closed surfaces, an elementary proof and extension of Minkowski's theorem, Riemannian geometry and a method for Dirichlet problems. Volume 2 covers general propositions on an intrinsic metric, angles and curvature, the existence of a convex polyhedron with prescribed metric, curves on convex surfaces, and the role of specific curvature. These monographs, published in English for the first time, will appeal to advanced students and researchers in mathematics and geometry.
Table of Contents
Volume 1: On infinitesimal bendings of nonregular surfaces. An elementary proof of the Minkowski and some other theorems on convex polyhedra. To the theory of mixed volumes of convex bodies. Part I: Extension of certain concepts of the theory of convex bodies. To the theory of mixed volumes of convex bodies. Part II: New inequalities for mixed volumes and their applications. To the theory of mixed volumes of convex bodies. Part III: Extension of two Minkowski theorems on convex polyhedra to all convex bodies. To the theory of mixed volumes of convex bodies. Part IV: Mixed discriminants and mixed volumes. A general uniqueness theorem for closed surfaces. On the area function of a convex body. Intrinsic geometry of an arbitrary convex surface. Existence of a convex polyhedron and a convex surface with given metric. On tiling a space with polyhedra. On a generalization of Riemannian geometry. The Dirichlet problem. A general method for dominating solutions of the Dirichlet problem. On the principles of relativity theory. Index. Volume 2: Basic Concepts and Results. General Propositions about the Intrinsic Metric. Characteristic Properties of the Intrinsic Metric. Angle. Curvature. Existence of a Convex Polyhedron with a Given Metric. Existence of a Closed Convex Surface with a Given Metric. Other Existence Theorems. Curves on Convex Surfaces. Area. The Role of the Specific Curvature. Generalization. Appendix. Basic Facts of Convex Bodies.