I. Convexity -- ยง1 Convex sets -- ยง2 Faces and supporting hyperplanes -- ยง3 Polarity -- ยง4 Direct sum decompositions -- ยง5 The lower semicontinuity of the operator โ{128}{156}expโ{128}{157} -- ยง6 Convex cones -- ยง7 The Farkas Lemma and its generalization -- ยง8 Separable systems of convex cones -- II. d-Convexity in normed spaces -- ยง9 The definition of d-convex sets -- ยง10 Support properties of d-convex sets -- ยง11 Properties of d-convex flats -- ยง12 The join of normed spaces -- ยง13 Separability of d-convex sets -- ยง14 The Helly dimension of a set family -- ยง15 d-Star-shaped sets -- III. H-convexity -- ยง16 The functional md for vector systems -- ยง17 The ?-displacement Theorem -- ยง18 Lower semicontinuity of the functional md -- ยง19 The definition of H-convex sets -- ยง20 Upper semicontinuity of the H-convex hull -- ยง21 Supporting cones of H-convex bodies -- ยง22 The Helly Theorem for H-convex sets -- ยง23 Some applications of H-convexity -- ยง24 Some remarks on connection between d-convexity and H-convexity -- IV. The Szรถkefalvi-Nagy Problem -- ยง25 The Theorem of Szรถkefalvi-Nagy and its generalization -- ยง26 Description of vector systems with md H = 2 that are not one-sided -- ยง27 The 2-systems without particular vectors -- ยง28 The 2-system with particular vectors -- ยง29 The compact, convex bodies with md M = 2 -- ยง30 Centrally symmetric bodies -- V. Borsukโ{128}{153}s partition problem -- ยง31 Formulation of the problem and a survey of results -- ยง32 Bodies of constant width in Euclidean and normed spaces -- ยง33 Borsukโ{128}{153}s problem in normed spaces -- VI. Homothetic covering and illumination -- ยง34 The main problem and a survey of results -- ยง35 The hypothesis of Gohberg-Markus-Hadwiger -- ยง36 The infinite values of the functional b, b2032;, c, c2032;, -- ยง37 Inner illumination of convex bodies -- ยง38 Estimates for the value of the functional p(K) -- VII. Combinatorial geometry of belt bodies -- ยง39 The integral respresentation of zonoids -- ยง40 Belt vectors of a compact, convex body -- ยง41 Definition of belt bodies -- ยง42 Solution of the illumination problem for belt bodies -- ยง43 Solution of the Szรถkefalvi-Nagy problem for belt bodies -- ยง44 Minimal fixing systems -- VIII. Some research problems -- Author Index -- List of Symbols