Excursions into Combinatorial Geometry

1997
ISBN: 978-3-540-61341-1
Verlag: Springer Berlin Heidelberg

Buch, Englisch, 423 Seiten, Paperback, Format (B × H): 155 mm x 235 mm, Gewicht: 1370 g

Reihe: Universitext

Geometry undoubtedly plays a central role in modern mathematics. And it is not only a physiological fact that 80 % of the information obtained by a human is absorbed through the eyes. It is easier to grasp mathematical con­ cepts and ideas visually than merely to read written symbols and formulae. Without a clear geometric perception of an analytical mathematical problem our intuitive understanding is restricted, while a geometric interpretation points us towards ways of investigation. Minkowski's convexity theory (including support functions, mixed volu­ mes, finite-dimensional normed spaces etc.) was considered by several mathe­ maticians to be an excellent and elegant, but useless mathematical device. Nearly a century later, geometric convexity became one of the major tools of modern applied mathematics. Researchers in functional analysis, mathe­ matical economics, optimization, game theory and many other branches of our field try to gain a clear geometric idea, before they start to work with formulae, integrals, inequalities and so on. For examples in this direction, we refer to [MalJ and [B-M 2J. Combinatorial geometry emerged this century. Its major lines of investi­ gation, results and methods were developed in the last decades, based on seminal contributions by O. Helly, K. Borsuk, P. Erdos, H. Hadwiger, L. Fe­ jes T6th, V. Klee, B. Griinbaum and many other excellent mathematicians.

Zielgruppe

Research

Autoren/Hrsg.

Boltyanski, Vladimir

Soltan, P. S.

Martini, Horst

Fachgebiete

• Mathematik | Informatik Mathematik Operations Research Spieltheorie
• Mathematik | Informatik Mathematik Topologie Mengentheoretische Topologie
• Mathematik | Informatik Mathematik Mathematische Analysis Variationsrechnung
• Mathematik | Informatik Mathematik Operations Research Graphentheorie
• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie

Inhaltsverzeichnis

I. Convexity.- §1 Convex sets.- §2 Faces and supporting hyperplanes.- §3 Polarity.- §4 Direct sum decompositions.- §5 The lower semicontinuity of the operator “exp”.- §6 Convex cones.- §7 The Farkas Lemma and its generalization.- §8 Separable systems of convex cones.- II. d-Convexity in normed spaces.- §9 The definition of d-convex sets.- §10 Support properties of d-convex sets.- §11 Properties of d-convex flats.- §12 The join of normed spaces.- §13 Separability of d-convex sets.- §14 The Helly dimension of a set family.- §15 d-Star-shaped sets.- III. H-convexity.- §16 The functional md for vector systems.- §17 The ?-displacement Theorem.- §18 Lower semicontinuity of the functional md.- §19 The definition of H-convex sets.- §20 Upper semicontinuity of the H-convex hull.- §21 Supporting cones of H-convex bodies.- §22 The Helly Theorem for H-convex sets.- §23 Some applications of H-convexity.- §24 Some remarks on connection between d-convexity and H-convexity.- IV. The Szökefalvi-Nagy Problem.- §25 The Theorem of Szökefalvi-Nagy and its generalization.- §26 Description of vector systems with md H = 2 that are not one-sided.- §27 The 2-systems without particular vectors.- §28 The 2-system with particular vectors.- §29 The compact, convex bodies with md M = 2.- §30 Centrally symmetric bodies.- V. Borsuk’s partition problem.- §31 Formulation of the problem and a survey of results.- §32 Bodies of constant width in Euclidean and normed spaces.- §33 Borsuk’s problem in normed spaces.- VI. Homothetic covering and illumination.- §34 The main problem and a survey of results.- §35 The hypothesis of Gohberg-Markus-Hadwiger.- §36 The infinite values of the functional b, b2032;, c, c2032;,.- §37 Inner illumination of convex bodies.- §38Estimates for the value of the functional p(K).- VII. Combinatorial geometry of belt bodies.- §39 The integral respresentation of zonoids.- §40 Belt vectors of a compact, convex body.- §41 Definition of belt bodies.- §42 Solution of the illumination problem for belt bodies.- §43 Solution of the Szökefalvi-Nagy problem for belt bodies.- §44 Minimal fixing systems.- VIII. Some research problems.- Author Index.- List of Symbols.