Książki

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.

Excursions into Combinatorial Geometry

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 II H-convexity
16 The functional md for vector systems
17 TheE-displacement Theorem
18 Lower semicontinuity of the functional md
19 The definition of H-convex-sets
20 Upper semicontinuity of 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 wiht 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 functionals b, b', c, c'
37 Inner illumination of convex bodies
38 Estimates for the value of the functional p(K) VII Cominatorial geometry of belt bodies
39 The integral representation of zonoids
40 Beltvectors of a compact, convex body
41 Definition of belt bodies
42 Solution of the illuminations problem for belt bodies
43 Solution of the Szökefalvi-Nagy problem for belt bodies
44 Minumal fixing systems VIII Some research problems Bibliography Author Index Subject Index List of Symbols