Excursions into Combinatorial Geometry

ISBN-13: 9783540613411 / Angielski / Miękka / 1996 / 423 str.

Vladimir Boltyanski;Horst Martini;P.S. Soltan

Opis

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 problems from the combinatorial geometry of convex bodies (unsolved in the general case): 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. Each section is supplemented by a wide range of exercises and the geometric approach to many topics is illustrated with the help of more than 250 figures.

Spis treści

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