Excursions into Combinatorial Geometry

The book deals with the combinatorial geometry of convex bodies in finite-dimensional spaces. A general introduction to geometric convexity is followed by the investigation of d-convexity and H-convexity, and by various applications. Recent research is discussed, for example the three (generally unsolved) problems from the combinatorial geometry of convex bodies: the Szoekefalvi-Nagy problem, the Borsuk problem, the Hadwiger covering problem. These and related questions are then applied to a new class of convex bodies which is a natural generalization of the class of zonoids: the class of belt bodies. Finally open research problems are discussed.

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.