Pak I. Lectures on Discrete and Polyhedral Geometry

Los Angeles: UCLA, 2010. — 442 p.The subject of Discrete Geometry and Convex Polytopes has received much attention in recent decades, with an explosion of the work in the field. This book is an introduction, covering some familiar and popular topics as well as some old, forgotten, sometimes obscure, and at times very recent and exciting results. It is somewhat biased by my personal likes and dislikes, and by no means is a comprehensive or traditional introduction to the field, as we further explain below. This book began as informal lecture notes of the course I taught at MIT in the Spring of 2005 and again in the Fall of 2006. The richness of the material as well as its relative inaccessibility from other sources led to making a substantial expansion. Also, the presentation is now largely self-contained, at least as much as we could possibly make it so. Let me emphasize that this is neither a research monograph nor a comprehensive survey of results in the field. The exposition is at times completely elementary and at times somewhat informal. Some additional material is included in the appendix and spread out in a number of exercises. The book is divided into two parts. The first part covers a number of basic results in discrete geometry and with few exceptions the results are easily available elsewhere (to a committed reader). The sections in the first part are only loosely related to each other. In fact, many of these sections are subjects of separate monographs, from which we at times borrow the proof ideas (see reference subsections for the acknowledgements). However, in virtually all cases the exposition has been significantly altered to unify and simplify the presentation. In and by itself the first part can serve as a material for the first course in discrete geometry, with fairly large breadth and relatively little depth (see more on this below). The second part is more coherent and can be roughly described as the discrete differential geometry of curves and surfaces. This material is much less readily available, often completely absent in research monographs, and, on more than one occasion, in the English language literature. We start with discrete curves and then proceed to discuss several versions of the Cauchy rigidity theorem, the solution of the bellows conjecture and Alexandrov’s various theorems on polyhedral surfaces. Although we do not aim to be comprehensive, the second part is meant to be as an introduction to polyhedral geometry, and can serve as a material for a topics class on the subject. Although the results in the first part are sporadically used in the second part, most results are largely independent. However, the second part requires a certain level of maturity and should work well as the second semester continuation of the first part. We include a large number of exercises which serve the dual role of possible home assignment and additional material on the subject. For most exercises, we either include a hint or a complete solution, or the references. The appendix is a small collection of standard technical results, which are largely available elsewhere and included here to make the book self-contained.Introduction.

Basic definitions and notations.
Basic discrete geometry
The Helly theorem.
Caratheodory and Barany theorems.
The Borsuk conjecture.
Fair division.
Inscribed and circumscribed polygons.
Dyson and Kakutani theorems.
Geometric inequalities.
Combinatorics of convex polytopes.
Center of mass, billiards and the variational principle.
Geodesics and quasi-geodesics.
The Steinitz theorem and its extensions.
Universality of point and line configurations.
Universality of linkages.
Triangulations.
Hilbert’s third problem.
Polytope algebra.
Dissections and valuations.
Monge problem for polytopes.
Regular polytopes.
Kissing numbers.
Discrete geometry of curves and surfaces
The four vertex theorem.
Relative geometry of convex polygons.
Global invariants of curves.
Geometry of space curves.
Geometry of convex polyhedra: basic results.
Cauchy theorem: the statement, the proof and the story.
Cauchy theorem: extensions and generalizations.
Mean curvature and Pogorelov’s lemma.
Senkin-Zalgaller’s proof of the Cauchy theorem.
Flexible polyhedra.
The algebraic approach.
Static rigidity.
Infinitesimal rigidity.
Proof of the bellows conjecture.
The Alexandrov curvature theorem.
The Minkowski theorem.
The Alexandrov existence theorem.
Bendable surfaces.
Volume change under bending.
Foldings and unfoldings.
Details, details...
Appendix.
Additional problems and exercises.
Hints, solutions and references to selected exercises.

Notation for the references.
Index.True PDF