CONVEX POLYHEDRA

By P. FANIE

CONTENT AND PURPOSE OF THE BOOK ORDER AND CHARACTER OF THE EXPOSITION REMARKS FOR THE PROFESSIONAL BASIC CONCEPTS AND SIMPLEST PROPERTIES OF CONVEX POLYHEDRA DE?NITION OF A CONVEX POLYHEDRON DETERMINING A POLYHEDRON FROM THE PLANES OF ITS FACES DETERMINING A CLOSED POLYHEDRON FROM ITS VERTICES DETERMINING AN UNBOUNDED POLYHEDRON FROM ITS VERTICES AND THE LIMIT ANGLE THE SPHERICAL IMAGE DEVELOPMENT TOPOLOGICAL PROPERTIES OF POLYHEDRA AND DEVELOPMENTS SOME THEOREMS OF THE INTRINSIC GEOMETRY OF DEVELOPMENTS GENERALIZATIONS METHODS AND RESULTS THE CAUCHY LEMMA THE MAPPING LEMMA DETERMINING A POLYHEDRON FROM A DEVELOPMENT (SURVEY OF CHAPTERS 3, 4, AND 5) POLYHEDRA WITH PRESCRIBED FACE DIRECTIONS (SURVEY OF CHAPTERS 6, 7, AND 8) POLYHEDRA WITH VERTICES ON PRESCRIBED RAYS (SURVEY OF CHAPTER 9) IN?NITESIMAL RIGIDITY THEOREMS (SURVEY OF CHAPTERS AND 11) PASSAGE FROM POLYHEDRA TO CURVED SURFACES BASIC TOPOLOGICAL NOTIONS THE DOMAIN INVARIANCE THEOREM UNIQUENESS OF POLYHEDRA WITH PRESCRIBED DEVELOPMENT SEVERAL LEMMAS ON POLYHEDRAL ANGLES EQUALITY OF DIHEDRAL ANGLES IN POLYHEDRA WITH EQUAL PLANAR ANGLES UNBOUNDED POLYHEDRA OF CURVATURE LESS THAN 2? POLYHEDRA WITH BOUNDARY EXISTENCE OF POLYHEDRA WITH PRESCRIBED DEVELOPMENT THE MANIFOLD OF DEVELOPMENTS THE MANIFOLD OF POLYHEDRA EXISTENCE OF CLOSED CONVEX POLYHEDRA WITH PRESCRIBED DEVELOPMENT EXISTENCE OF UNBOUNDED CONVEX POLYHEDRA WITH PRESCRIBED DEVELOPMENT EXISTENCE OF UNBOUNDED POLYHEDRA GIVEN THE DEVELOPMENT AND THE LIMIT ANGLE GLUING AND FLEXING POLYHEDRA WITH BOUNDARY GLUING POLYHEDRA WITH BOUNDARY FLEXES OF CONVEX POLYHEDRA GENERALIZATIONS OF CHAPTERS 4 AND 5 CONDITIONS FOR CONGRUENCE OF POLYHEDRA WITH PARALLEL FACES LEMMAS ON CONVEX POLYHEDRA ON LINEAR COMBINATION OF POLYHEDRA CONGRUENCE CONDITIONS FOR CLOSED POLYHEDRA CONGRUENCE CONDITIONS FOR UNBOUNDED POLYHEDRA ANOTHER PROOF AND GENERALIZATION OF THE THEOREM ON UNBOUNDED POLYHEDRA POLYHEDRA WITH BOUNDARY EXISTENCE THEOREMS FOR POLYHEDRA WITH PRESCRIBED FACE DIRECTIONS EXISTENCE OF POLYHEDRA WITH PRESCRIBED FACE AREAS MINKOWSKI’S PROOF OF THE EXISTENCE OF POLYHEDRA WITH PRESCRIBED FACE AREAS EXISTENCE OF UNBOUNDED POLYHEDRA WITH PRESCRIBED FACE AREAS THE GENERAL EXISTENCE THEOREM FOR UNBOUNDED POLYHEDRA EXISTENCE OF CONVEX POLYHEDRA WITH PRESCRIBED SUPPORT NUMBERS RELATIONSHIP BETWEEN THE CONGRUENCE CONDITION FOR POLYHEDRA WITH PARALLEL FACES AND OTHER PROBLEMS PARALLELOHEDRA A POLYHEDRON OF LEAST AREA WITH FIXED VOLUME MIXED VOLUMES AND THE BRUNN INEQUALITY POLYHEDRA WITH VERTICES ON PRESCRIBED RAYS CLOSED POLYHEDRA UNBOUNDED POLYHEDRA 10 IN?NITESIMAL RIGIDITY OF CONVEX POLYHEDRA WITH STATIONARY DEVELOPMENT 10 1 DEFORMATION OF POLYHEDRAL ANGLES 10 2 THE STRONG CAUCHY LEMMA 10 3 STATIONARY DIHEDRAL ANGLES FOR STATIONARY PLANAR ANGLES 10 4 IN?NITESIMAL RIGIDITY OF POLYHEDRA AND EQUILIBRIUM OF HINGE MECHANISMS 10 5 ON THE DEFORMATION OF DEVELOPMENTS 10 6 RIGIDITY OF POLYHEDRA WITH STATIONARY DEVELOPMENT 10 7 GENERALIZATIONS 11 IN?NITESIMAL RIGIDITY CONDITIONS FOR POLYHEDRA WITH PRESCRIBED FACE DIRECTIONS 11 1 ON DEFORMATIONS OF POLYGONS 11 2 IN?NITESIMAL RIGIDITY THEOREMS FOR POLYHEDRA 11 3 RELATIONSHIP OF IN?NITESIMAL RIGIDITY THEOREMS WITH ONE ANOTHER AND WITH THE THEORY OF MIXED VOLUMES 11 4 GENERALIZATIONS 12 SUPPLEMENTS 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 12 2 SUPPLEMENT TO CHAPTER 4: YU A VOLKOV EXISTENCE OF CONVEX POLYHEDRA WITH PRESCRIBED DEVELOPMENT I 12 3 SUPPLEMENT TO CHAPTER 5: L A SHOR ON FLEXIBILITY OF CONVEX POLYHEDRA WITH BOUNDARY REFERENCES