Guggenheimer H.W. Applicable Geometry: Global and Local Convexity

Huntington: Robert E. Krieger Publishing Company. – 1977. – 213 p. (Applied mathematics series) The Department of Mathematics of the Polytechnic Institute of New York sees its major task in the training of applied and industrial mathematicians. In order to allow an undergraduate two years to study a great variety of topics in applied mathematics and statistics, our preferred program provides in the first two years of study not only the usual four semester course in Calculus and Elementary Differential Equations but also one semester each of Programing, Linear Algebra, Abstract Algebra, and Geometry. The geometry course developed for this program consists of the major parts of chapters 1, 2, 4 of the present book together with the first part of section 8, of 3th chapter. These materials could be characterized as "What every Applied Analyst should know of Geometry" including those parts of analytic and differential geometry that are most frequently encountered in analytical applications. The main parts of sections 8 and 9 are usually included in my graduate course on Applied Ordinary Differential Equations. The stress in the text is naturally more on techniques than on results; an extensive collection of specific results is proved in the problems for which short outlines of proofs are given in the back of the book. A short first chapter contains the introductory material. The second chapter contains the basic theory of convex bodies: Minkowski addition and the Blaschke convergence theorems, support and distance functions, and duality. The main topic of the third chapter is local convexity, applied to ordinary second order differential equations. The fourth chapter centers around Minkowski's theory of mixed volumes in the form given to it by Hadwiger, applied to integral geometry, the basics of differential geometry of surfaces, and Helly's theorem.Preliminaries.
Convex bodies.
Ordinary differential equations.
Functions of convex bodies.Solutions to Exercises.