Convex Polyhedra

A.D. Alexandrov was awarded the Stalin State Prize in 1942, the Lobachevsky prize in 1952, and the Euler Golden Medal in 1992.

Basic Concepts and Simplest Properties of Convex Polyhedra.- Methods and Results.- Uniqueness of Polyhedra with Prescribed Development.- Existence of Polyhedra with Prescribed Development.- Gluing and Flexing Polyhedra with Boundary.- Congruence Conditions for Polyhedra with Parallel Faces.- Existence Theorems for Polyhedra with Prescribed Face Directions.- Relationship Between the Congruence Condition for Polyhedra with Parallel Faces and Other Problems.- Polyhedra with Vertices on Prescribed Rays.- Infinitesimal Rigidity of Convex Polyhedra with Stationary Development.- Infinitesimal Rigidity Conditions for Polyhedra with Prescribed Face Directions.- Supplements.

From the reviews:

"This book was first published in Russian in 1950 [2], then translated into German in 1958 [3]. Now, V. A. Zalgaller has updated and substantially lengthened this English edition with more recent related results and translations of hard-to-find related results. [...] For me, the star result in this book has to do with the realizability of developments of convex polyhedra. [...] [Alexandrov's existence theorem] is explained carefully in this book and is a substantial accomplishment. [...] In the present translation of A. D. Alexandrov's book, in addition to several footnotes concerning more recent results, Zalgaller has included supplements which are translations of papers of Yu. A. Volkov that explain a somewhat simpler and shorter (but still quite nontrivial) proof of Pogorelov's result above ... This gives the book a completeness and accessibility that has so far been sadly lacking, at least for English speakers in the West. [...] But Alexandrov's book has several interesting discussions of other closely related subjects in addition to the existence and uniqueness of polyhedra having given metrics. [...] Another interesting topic is Alexandrov's discussion of the realization of convex polyhedra whose vertices lie on rays from a point. There is also a very interesting discussion connecting the Cauchy-type problem with the Minkowski-type problem formally and showing how the Brunn- Minkowski inequalities can be brought to bear.I would definitely recommend this book to a student who would like to get acquainted with some of the ideas in this sort of discrete geometry. There are little goodies throughout that are very enlightening, and the discussion is very conversational [...]" (R. Connelly, Cornell University, SIAM Reviews, 48:1, 2006)

"The book is one of the classics in geometry. It covers a wealth of aspects of the theory of 3-dimensional convex polyhedra, nowhere to be found in any other book ina comparable way, and in anywhere near its detail and completeness. It contains the available answers to the question of the data uniquely determining a convex polyhedron. [...] The book includes the basics of convex polyhedra and collects the most general existence theorems for convex polyhedra that are proved by a new and unified method. It is a wonderful source of ideas for students. The English edition includes numerous comments as well as added material and a comprehensive bibliography by V.A. Zalgaller to bring the work up to date. Moreover, related papers by L.A. Shor and Yu.A. Volkov have been added as supplements to this book." (Zentralblatt für Didaktik der Mathematik, August 2005)

"Convex polyhedra presents a complete and detailed survey of the synthetic theory of convex polyhedra in three-dimensional Euclidean space. Clearly written, it may serve to graduate students and non-specialists as a comprehensive introduction to the subject. On the other hand, it still remains of great interest for specialists, since the present edition contains a series of new results, methods and open problems which are published for the first time." (Vasyl Gorkaviy, Zentralblatt MATH, 1067, 2005)

"The dominant theme of the book concerns the data required to specify a convex polyhedron uniquely ... this book contains much fascinating material ... the general approach may be such as to motivate interest in the newly emerging topic of discrete differential geometry." (P.N. Ruane, The Mathematical Gazette, 90:519, 2006)

"This classic text was published (in Russian) in 1950, and translated into German in 1958. It treats the metrical theory of ordinary convex polyhedra ... . this felicitous translation into English is very welcome. Notes have been added throughout ... . For every geometer's bookshelf, and of wide interest to the general mathematician." (Mathematika, Vol. 52, 2005)

"This book is a true classic, anda pleasure to read. It is devoted to the following question: Which data determine a (threedimensional) convex polyhedron and to what extent? The Russian edition appeared in 1950, a German translation in 1958. At last, this English translation makes it more easily accessible for a much wider readership." (P. Schmitt, Monatshefte für Mathematik, Vol. 151 (4), 2007)