Alexandr Danilovich Alexandrov has been called a giant of 20th-century mathematics. This volume contains some of the most important papers by this renowned geometer and hence, some of his most influential ideas. Alexandrov addressed a wide range of modern mathematical problems, and he did so with intelligence and elegance, solving some of the discipline's most difficult and enduring challenges. He was the first to apply many of the tools and methods of the theory of real functions and functional analysis that are now current in geometry. The topics here include convex polyhedrons and closed surfaces, an elementary proof and extension of Minkowski's theorem, Riemannian geometry and a method for Dirichlet problems. This monograph, published in English for the first time, gives unparalleled access to a brilliant mind, and advanced students and researchers in applied mathematics and geometry will find it indispensable.
Table of Contents
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.
Yu. G. Reshetnyak, S. S. Kutateladze