Bodies of Constant Width: An Introduction to Convex Geometry with Applications

Description

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. Front Matter ....Pages i-xi Introduction (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 1-12 Convex Geometry (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 13-58 Basic Properties of Bodies of Constant Width (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 59-74 Figures of Constant Width (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 75-93 Systems of Lines in the Plane (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 95-126 Spindle Convexity (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 127-142 Complete and Reduced Convex Bodies (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 143-165 Examples and Constructions (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 167-195 Sections of Bodies of Constant Width (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 197-207 Bodies of Constant Width in Minkowski Spaces (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 209-245 Bodies of Constant Width in Differential Geometry (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 247-277 Mixed Volumes (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 279-297 Bodies of Constant Width in Analysis (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 299-320 Geometric Inequalities (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 321-342 Bodies of Constant Width in Discrete Geometry (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 343-367 Bodies of Constant Width in Topology (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 369-398 Concepts Related to Constant Width (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 399-424 Bodies of Constant Width in Art, Design, and Engineering (Horst Martini, Luis Montejano, Déborah Oliveros)....Pages 425-443 Back Matter ....Pages 445-486