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 -- §38 Estimates 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.