## A. D. Alexandrov Selected Works: Selected Scientific PapersAlexandr 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. |

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### Contents

An elementary proof of the Minkowski and some | 19 |

To the theory of mixed volumes of convex bodies | 31 |

To the theory of mixed volumes of convex bodies | 99 |

To the theory of mixed volumes of convex bodies | 119 |

A general uniqueness theorem for closed surfaces | 145 |

On the area function of a convex body | 155 |

Intrinsic geometry of an arbitrary convex surface | 163 |

Existence of a convex polyhedron and a convex | 169 |

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A'-plane analogous arbitrary assertion assume bending field body H boundary bounded Brunn inequality completes the proof cones Cx Consequently continuous function convex bodies convex functions convex metric convex polyhedron convex surface corresponding triangle curve defined definition denote derivative differential disk domain G domain RK eigenvalue equality holds equation everywhere exists finite follows formula function Z(n geometry Hence Hn-i homeomorphic homothetic integral interior points intersection intrinsic metric inversion isometric isotropic line l)-dimensional faces Lemma length linear mapping metric space Minkowski mixed area function mixed discriminants mixed volumes obtain parallel polygon PI polyhedra polyhedron projection Proposition prove rectifiable curve segment sequence set function shortest arc sides sinh solution space of curvature space Rn spherical neighborhood sufficiently support function support numbers supporting planes surface of revolution tends to zero theorem transformation translation triangle ABC triangle TK unique unit ball unit sphere upper angle vanishes vector vertex vertices virtue