213:. A generalization of this theorem implies that the same is true for the perimeters and directions of the faces. Chapter 9 concerns the reconstruction of three-dimensional polyhedra from a two-dimensional perspective view, by constraining the vertices of the polyhedron to lie on rays through the point of view. The original Russian edition of the book concludes with two chapters, 10 and 11, related to
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to students and professional mathematicians as an introduction to the mathematics of convex polyhedra. He also writes that, over 50 years after its original publication, "it still remains of great interest for specialists", after being updated to include many new developments and to list new open
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was long overdue. He calls the material on
Alexandrov's uniqueness theorem "the star result in the book", and he writes that the book "had a great influence on countless Russian mathematicians". Nevertheless, he complains about the book's small number of exercises, and about an inconsistent level
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The 2005 English edition adds comments and bibliographic information regarding many problems that were posed as open in the 1950 edition but subsequently solved. It also includes in a chapter of supplementary material the translations of three related articles by Volkov and Shor, including a
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The main focus of the book is on the specification of geometric data that will determine uniquely the shape of a three-dimensional convex polyhedron, up to some class of geometric transformations such as congruence or similarity. It considers both bounded polyhedra
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on the impossibility of labeling the edges of a polyhedron by positive and negative signs so that each vertex has at least four sign changes, the remainder of chapter 2 outlines the content of the remaining book. Chapters 3 and 4 prove
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The 1950 Russian edition of the book included 11 chapters. The first chapter covers the basic topological properties of polyhedra, including their topological equivalence to spheres (in the bounded case) and
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writes that, for a work describing significant developments in the theory of convex polyhedra that was however hard to access in the west, the
English translation of
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194:. Chapter 5 considers the metric spaces defined in the same way that are topologically a disk rather than a sphere, and studies the
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simplified proof of
Pogorelov's theorems generalizing Alexandrov's uniqueness theorem to non-polyhedral convex surfaces.
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presentation that fails to distinguish important and basic results from specialized technicalities.
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a convex polyhedron is uniquely determined by the areas and directions of its faces
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of finite sets of points) and unbounded polyhedra (intersections of finitely many
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in 1958. An updated edition, translated into
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assumes a significant level of background knowledge in material including
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Chapters 6 through 8 of the book are related to a theorem of
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1950 book on geometry by
Aleksandr Danilovich Aleksandrov
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Although intended for a broad mathematical audience,
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159:that are topologically spherical locally like the
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95:Aleksandr Danilovich Aleksandrov
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269:problems in the area.
233:Audience and reception
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99:Выпуклые многогранники
39:Выпуклые многогранники
258:differential geometry
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187:{\displaystyle 4\pi }
48:Nurlan S. Dairbekov,
320:Mathematical Reviews
211:invariance of domain
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146:. After a lemma of
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203:Hermann Minkowski
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68:Mathematics
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285:References
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45:Translator
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273:See also
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223:Max Dehn
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60:Russian
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