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This classic geometry text explores the theory of 3-dimensional convex polyhedra in a unique fashion, with exceptional detail. Vital and clearly written, 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. This edition includes a comprehensive bibliography by V.A. Zalgaller, and related papers as supplements to the original text.
Convex Polyhedra is one of the classics in geometry. There simply is no other book with so many of the aspects of the theory of 3-dimensional convex polyhedra in a comparable way, and in anywhere near its detail and completeness. It is the definitive source of the classical field of convex polyhedra and contains the available answers to the question of the data uniquely determining a convex polyhedron. This question concerns all data pertinent to the poly hedron, e.g. the lengths of edges, areas of faces, etc. This viatal and clearly written 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.
Excerpt from On Shortest Paths Amidst Convex Polyhedra Let K be a 3-D convex polyhedron having n vertices. A sequence of edges of K is called a shortest-path sequence if there exist two points X, Y on the surface S of K such that is the sequence of edges crossed by the shortest path from X to Y along S. We show that the number of shortest-path sequences for K is polynomial inn, and as a consequence prove that the shortest path between two points in 3-space which must avoid the interiors of a fixed number of disjoint convex polyhedral obstacles, can be calculated in time polynomial in the total number of vertices of these obstacles (but exponential in the number of obstacles). 1. Introduction In this paper we study several problems related to the problem of calculating the Euclidean shortest path between two points in 3-dimensional space, which must avoid the interiors of a collection of polyhedral obstacles having altogether n vertices. This general problem seems to be intractable, and the only known algorithms for it require exponential time ([SS], [RS]), although no lower bounds are known as yet for this problem. On the other extreme hand we have the problem of finding the shortest path between two points in 3-space which must avoid the interior of a single convex polyhedral obstacle. In this case the problem is solvable in time 0(n log n) ([SS], [Mo]). Interpolating between these two extreme cases, one might consider the problem in which the polyhedral obstacles consist of a fixed number k of disjoint convex polyhedra (having altogether n vertices), and attempt to calibrate the complexity of this problem as a function of k and n. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works."
This is a self-contained exposition of several core aspects of the theory of rational polyhedra with a view towards algorithmic applications to efficient counting of integer points, a problem arising in many areas of pure and applied mathematics. The approach is based on the consistent development and application of the apparatus of generating functions and the algebra of polyhedra. Topics range from classical, such as the Euler characteristic, continued fractions, Ehrhart polynomial, Minkowski Convex Body Theorem, and the Lenstra-Lenstra-Lovasz lattice reduction algorithm, to recent advances such as the Berline-Vergne local formula. The text is intended for graduate students and researchers. Prerequisites are a modest background in linear algebra and analysis as well as some general mathematical maturity. Numerous figures, exercises of varying degree of difficulty as well as references to the literature and publicly available software make the text suitable for a graduate course.