Convex Figures and Polyhedra – Lyusternik

In this post, we will see the book Convex Figures And Polyhedra by L. A. Lyusternik. This book is a part of Topics in Mathematics series.

About the book

The theory of convex figures and polyhedra provides an excellent example of a body of mathematical knowledge that offers theorems with elementary formulations and vivid geometric meaning. Despite this simplicity of formulation, the proofs are often not elementary. Thus the area presents a particular challenge to mathematicians, who have investigated convex figures and polyhedra for millenia, and yet have by far not exhausted the subject. Many of the theorems in this volume were in fact proved only a few years ago.

The material in this book will be suitable for study in mathe­matics clubs or by readers with a background of secondary school mathematics only. The topics considered are stimulating and chal­ lenging, and moreover, convexity ideas are valuable in the study of modem higher mathematics. Mathematical analysis, higher geome­ try, and topology each use convexity notions in an essential way.

The book was translated from Russian by Donald L. Barnett and was published in 1966.

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CHAPTER l. Convex Figures 1

1. Plane convex figures 1
2. Intersections and partitions of plane convex figures 5
3. Supporting lines for two-dimensional convex figures 8
4. Directed convex curves and directed supporting lines 10
5. Vectors; external normals to plane convex figures 13
6. Circuit of a polygon; length of a convex curve 14
7. Convex solids 17
8. Supporting planes and external normals for convex solids 20
9. Central projection; cones 23
10. Convex spherical figures 26
11. Greatest and least widths of convex figures 28
12. Ovals of constant width; Barbier’s Theorem 34

CHAPTER 2. Central-symmetric Convex Figures 39

13. Central symmetry and (parallel) translation 39
14. Partitioning central-symmetric polyhedra 42
15. The greatest central-symmetric convex figure in a lattice of integers; Minkowski’s Theorem 44
16. Filling the plane and space with convex figures 51

CHAPTER 3. Networks and Convex Polyhedra 58

17. Vertices (nodes), faces (regions), and edges (lines); Euler’s Theorem 58
18. Proof of the theorem for connected networks 61
19. Disconnected networks; inequalities 64
20. Congruent and symmetric polyhedra; Cauchy’s Theorem 66
21. Proof of Cauchy’s Theorem 71
22. Steinitz’ correction of Cauchy’s proof 73
23. Abstract and convex polyhedra; Steinitz Theorem 81
24. Development of a convex polyhedron; Aleksandrov’s Theorem 95

CHAPTER 4. Linear Systems of Convex Figures 97

25. Linear operations on points

26. Linear operations on figures; “mixing” figures

27. Linear systems of convex polygons; areas and “mixed areas”
28. Applications
29. Schwarz inequality; other inequalities
30. Relation between areas of Q, Q_{1}, and Q_{S},; the Brunn-Minkowski inequality
31. Relation between areas of plane sections of convex solids
32. Greatest area theorems

CHAPTER 5. Theorems of Minkowski and Aleksandrov for Congruent Convex Polyhedra 132

33. Formulation of the theorems 132
34. A theorem about convex polygons 134
35. Mean polygons and polyhedra 141
36. Proof of Aleksandrov’s Theorem 146

CHAPTER 6. Supplement 150

37. Precise definition of a convex figure 150
38. Continuous mapping and functions 152
39. Regular networks; regular and semiregular polyhedra 153
40. The isoperimetric problem 164
41. Chords of arbitrary continua; Levi’s Theorem 166
42. Figures in a lattice of integers; Blichfeldt’s Theorem 172
43. Topological theorems of Lebesgue and Bol’-Brouwer 175
44. Generalization to n dimensions 182
45. Convex figures in normed spaces 185

Bibliography 191


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