1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

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Convex bodies : the Brunn-Minkowski theory

From the Publisher:
Discrete geometry investigates combinatorial properties of configurations of geometric objects. To a working mathematician or computer scientist, it offers sophisticated results and techniques of great diversity and it is a foundation for fields such as computational geometry or combinatorial optimization.
This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and
high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces.
Jiri Matousek is Professor of Computer Science at Charles University in Prague. His research has contributed to several of the considered areas and to their algorithmic applications. This is his third book.

Lectures on Discrete Geometry

Convex bodies: the brunn–minkowski theory

A number of authors have defined contractive type mappings on a complete metric space X which are generalizations of the well-known Banach contraction, and which have the property that each such mapping has a unique fixed point. The fixed point can always be found by using Picard iteration, beginning with some initial choice xo E X. In this paper we compare this multitude of definitions. X denotes a complete metric space with distance function d, and f a function mapping X into itself. 1. Definitions of contractive type mappings. (1) (Banach) There exists a number a, 0 6 a < 1, such that, for each x,y E X, d(f (x),f (y)) < ad(xY)(2) (Rakotch [21]) There exists a monotone decreasing function a: (0, oo) [0, 1) such that, for each x, y E= X, x # y, d(f (x)j(y)) <ad(x,Y)(3) (Edelstein [10]) For each x, y E X, x # y, d(f (x),f (y)) < d(x,y). (4) (Kannan [18]) There exists a number a, 0 < a < !, such that, for each x,y E X, d(f (x),f (y)) < a[d(x,f (x)) + d(y,f (y))]. (5) (Bianchini [3]) There exists a number h, 0 < h < 1, such that, for each x,,y E X, d(f (x),f (y)) < h max{d(x,f (x)), d(y,f (y))} (6) For each x, y E X, x # y, Received by the editors June 4, 1974 and, in revised form, March 5, 1975 and November 18, 1975. AMS (MOS) subject classifications (1970). Primary 47H10. c American Maithematical Society 1977 257 This content downloaded from 157.55.39.17 on Fri, 02 Sep 2016 04:42:04 UTC All use subject to http://about.jstor.org/terms

/pdf/a-comparison-of-various-definitions-of-contractive-mappings-502ily02qf.pdf

A comparison of various definitions of contractive mappings

Pre-Requisites of Fixed Points.- The Picard Iteration.- The Krasnoselskij Iteration.- The Mann Iteration.- The Ishikawa Iteration.- Other Fixed Point Iteration Procedures.- Stability of Fixed Point Iteration Procedures.- Iterative Solution of Nonlinear Operator Equations.- Error Analysis of Fixed Point Iteration Procedures.

http://www.gbv.de/dms/goettingen/526955376.pdf

Iterative Approximation of Fixed Points

Fix point theorems in metric spaces

A two-connected planar graph without concurrent longest paths

Hamiltonian properties of Toeplitz graphs

Most Monotone Functions are Singular

in the sense of "all, except those in a set of first Baire category". The space of all convex surfaces in IR", endowed with Hausdorfl's metric, is a Baire space. We shall see how abnormal convex surfaces may be, by proving that most of them are so.

Tudor Zamfirescu

Papers

Fix point theorems in metric spaces

A two-connected planar graph without concurrent longest paths

Hamiltonian properties of Toeplitz graphs

Most Monotone Functions are Singular

Many Endpoints and Few Interior Points of Geodesics