Convex polyhedra

Cover......Page 1
Title: Convex Polyhedra by A. D. Alexandrov......Page 3
© Springer-Verlag Berlin Heidelberg 2005......Page 4
Preface to the English Translation......Page 6
Contents......Page 7
Content and Purpose of the Book......Page 10
Order and Character of the Exposition......Page 12
Remarks for the Professional......Page 14
1.1 Definition of a Convex Polyhedron......Page 16
1.2 Determining a Polyhedron from the Planes of Its Faces......Page 25
1.3 Determining a Closed Polyhedron from Its Vertices......Page 30
1.4 Determining an Unbounded Polyhedron from Its Vertices and the Limit Angle......Page 35
1.5 The Spherical Image......Page 48
1.6 Development......Page 58
1.7 Topological Properties of Polyhedra and Developments......Page 65
1.8 Some Theorems of the Intrinsic Geometry of Developments......Page 81
1.9 Generalizations......Page 91
2.1 The Cauchy Lemma......Page 96
2.2 The Mapping Lemma......Page 102
2.3 Determining a Polyhedron from a Development (Survey of Chapters 3, 4, and 5)......Page 108
2.4 Polyhedra with Prescribed Face Directions (Survey of Chapters 6, 7, and 8)......Page 115
2.5 Polyhedra with Vertices on Prescribed Rays (Survey of Chapter 9)......Page 126
2.6 Infinitesimal Rigidity Theorems (Survey of Chapters 10 and 11)......Page 133
2.7 Passage from Polyhedra to Curved Surfaces......Page 145
2.8 Basic Topological Notions......Page 150
2.9 The Domain Invariance Theorem......Page 156
3.1 Several Lemmas on Polyhedral Angles......Page 164
3.2 Equality of Dihedral Angles in Polyhedra with Equal Planar Angles......Page 172
3.3 Uniqueness of Polyhedra with Prescribed Development......Page 178
3.4 Unbounded Polyhedra of Curvature Less Than 2π......Page 182
3.5 Polyhedra with Boundary......Page 191
3.6 Generalizations......Page 195
4.1 The Manifold of Developments......Page 202
4.2 The Manifold of Polyhedra......Page 211
4.3 Existence of Closed Convex Polyhedra with Prescribed Development......Page 219
4.4 Existence of Unbounded Convex Polyhedra with Prescribed Development......Page 221
4.5 Existence of Unbounded Polyhedra Given the Development and the Limit Angle......Page 227
5.1 Gluing Polyhedra with Boundary......Page 238
5.2 Flexes of Convex Polyhedra......Page 255
5.3 Generalizations of Chapters 4 and 5......Page 270
6.1 Lemmas on Convex Polyhedra......Page 280
6.2 On Linear Combination of Polyhedra......Page 290
6.3 Congruence Conditions for Closed Polyhedra......Page 296
6.4 Congruence Conditions for Unbounded Polyhedra......Page 300
6.5 Another Proof and Generalization of the Theorem on Unbounded Polyhedra. Polyhedra with Boundary......Page 304
6.6 Generalizations......Page 313
7.1 Existence of Polyhedra with Prescribed Face Areas......Page 320
7.2 Minkowski’s Proof of the Existence of Polyhedra with Prescribed Face Areas......Page 326
7.3 Existence of Unbounded Polyhedra with Prescribed Face Areas......Page 330
7.4 The General Existence Theorem for Unbounded Polyhedra......Page 336
7.5 Existence of Convex Polyhedra with Prescribed Support Numbers......Page 341
7.6 Generalizations......Page 351
8.1 Parallelohedra......Page 358
8.2 A Polyhedron of Least Area with Fixed Volume......Page 368
8.3 Mixed Volumes and the Brunn Inequality......Page 375
9.1 Closed Polyhedra......Page 386
9.2 Unbounded Polyhedra......Page 397
9.3 Generalizations......Page 404
10. Infinitesimal Rigidity of Convex Polyhedra with Stationary Development......Page 412
10.1 Deformation of Polyhedral Angles......Page 413
10.2 The Strong Cauchy Lemma......Page 419
10.3 Stationary Dihedral Angles for Stationary Planar Angles......Page 424
10.4 Infinitesimal Rigidity of Polyhedra and Equilibrium of Hinge Mechanisms......Page 430
10.5 On The Deformation of Developments......Page 434
10.6 Rigidity of Polyhedra with Stationary Development......Page 438
10.7 Generalizations......Page 444
11.1 On Deformations of Polygons......Page 448
11.2 Infinitesimal Rigidity Theorems for Polyhedra......Page 454
11.3 Relationship of Infinitesimal Rigidity Theorems with One Another and with the Theory of Mixed Volumes......Page 462
11.4 Generalizations......Page 468
12.1 Supplement To Chapter 3: Yu. A. Volkov. An Estimate for the Deformation of a Convex Surface in Dependence on the Variation of Its Intrinsic Metric......Page 472
12.2 Supplement to Chapter 4: Yu. A. Volkov. Existence of Convex Polyhedra with Prescribed Development. I......Page 501
1 Introduction. Statement of the Result......Page 515
2 Auxiliary Propositions......Page 519
3 Rigidity of Polyhedra of the Class K3......Page 521
4 Rigidity of Polyhedra of the Class K2......Page 526
5 Flexibility of Polyhedra Not Belonging to the Classes K1, K2,and K3......Page 532
References......Page 534
Subject Index......Page 546