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© 2019

Bodies of Constant Width

An Introduction to Convex Geometry with Applications

Textbook

Table of contents

  1. Front Matter
    Pages i-xi
  2. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 1-12
  3. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 13-58
  4. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 59-74
  5. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 75-93
  6. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 95-126
  7. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 127-142
  8. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 143-165
  9. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 167-195
  10. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 197-207
  11. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 209-245
  12. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 247-277
  13. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 279-297
  14. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 299-320
  15. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 321-342
  16. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 343-367
  17. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 369-398
  18. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 399-424
  19. Horst Martini, Luis Montejano, Déborah Oliveros
    Pages 425-443
  20. Back Matter
    Pages 445-486

About this book

Introduction

This is the first comprehensive monograph to thoroughly investigate constant width bodies, which is a classic area of interest within convex geometry. It examines bodies of constant width from several points of view, and, in doing so, shows surprising connections between various areas of mathematics. Concise explanations and detailed proofs demonstrate the many interesting properties and applications of these bodies. Numerous instructive diagrams are provided throughout to illustrate these concepts.

An introduction to convexity theory is first provided, and the basic properties of constant width bodies are then presented. The book then delves into a number of related topics, which include
 
  • Constant width bodies in convexity (sections and projections, complete and reduced sets, mixed volumes, and further partial fields)
  • Sets of constant width in non-Euclidean geometries (in real Banach spaces, and in hyperbolic, spherical, and further non-Euclidean spaces)
  • The concept of constant width in analysis (using Fourier series, spherical integration, and other related methods)
  • Sets of constant width in differential geometry (using systems of lines and discussing notions like curvature, evolutes, etc.)
  • Bodies of constant width in topology (hyperspaces, transnormal manifolds, fiber bundles, and related topics)
  • The notion of constant width in discrete geometry (referring to geometric inequalities, packings and coverings, etc.)
  • Technical applications, such as film projectors, the square-hole drill, and rotary engines

Bodies of Constant Width: An Introduction to Convex Geometry with Applications will be a valuable resource for graduate and advanced undergraduate students studying convex geometry and related fields. Additionally, it will appeal to any mathematicians with a general interest in geometry.


Keywords

Constant Width Bodies Convex Geometry Convexity Meissner bodies Reuleaux polygons Mixed Volumes Spherical Integration Combinatorial aspects of convexity Convex polyhedra Orbiforms

Authors and affiliations

  1. 1.Faculty of MathematicsChemnitz University of TechnologyChemnitzGermany
  2. 2.Instituto de MatemáticasUniversidad Nacional Autónoma de México, Campus JuriquillaQuerétaroMexico
  3. 3.Instituto de MatemáticasUniversidad Nacional Autónoma de México, Campus JuriquillaQuerétaroMexico

About the authors

Horst Martini obtained his PhD in Dresden/Germany (1984) and defended his habilitation at the Schiller University in Jena (1988). Since 1993 he is Full Professor for Mathematics at the University of Technology in Chemnitz, Germany (Chair of Geometry). He received a von Humboldt Scholarship in 1990, and an honorary professorship from the Harbin University of Science and Technology (China) in 2015. His main mathematical interests are focused on convex and discrete geometry, the geometry of finite dimensional real Banach spaces, and combinatorics.

Luis Montejano had graduate studies at MIT and the University of Utah, and he received his PhD in 1980. As a postdoc, he was member of the Institute for Advanced Studies, Princeton, and in 1990 he obtained a von Humboldt Scholarship in Heidelberg. He is Professor at the National University of Mexico, Campus Querétaro. His main mathematical interests refer to applications of topology to discrete and convex geometry.

Deborah Oliveros studied mathematics at the National University of Mexico UNAM, and she obtained her PhD in 1997. After that she got a postdoctoral position at the University of Calgary, Canada, where she stayed for several years. She is now Professor at the National University of Mexico, Campus Juriquilla, and her main mathematical interests are convex and abstract polytopes, discrete geometry, and convexity.

Bibliographic information