I Convexity Convex sets Faces and supporting hyperplanes Polarity Direct sum decompositions The lower semicontinuity of the operator 'exp' Convex cones The Farkas Lemma and its generalization Separable Systems of convex cones II d-Convexity in normed spaces The definition of d-convex sets Support properties of d-convex sets Properties of d-convex flats The join of normed spaces Separability of d-convex sets The Helly dimension of a set family d-Star-shaped sets II H-convexity The functional md for vector systems TheE-displacement Theorem Lower semicontinuity of the functional md The definition of H-convex-sets Upper semicontinuity of H-convex hull Supporting cones of H-convex bodies The Helly Theorem for H-convex sets Some applications of H-convexity Some remarks on connection between d-convexity and H-convexity IV The Szefalvi-Nagy Problem The Theorem of Szefalvi-Nagy and its generalization Description of vector systems with md H=2 that are not one-sided The 2-systems without particular vectors The 2-system wiht particular vectors The compact, convex bodies with md M=2 Centrally symmetric bodies V Borsuk's partition problem Formulation of the problem and a survey of results Bodies of constant width in Euclidean and normed spaces Borsuk's problem in normed spaces VI Homothetic covering and illumination The main problem and a survey of results The hypothesis of Gohberg-Markus-Hadwiger The infinite values of the functionals b, b', c, c' Inner illumination of convex bodies Estimates for the value of the functional p(K) VII Cominatorial geometry of belt bodies The integral representation of zonoids Beltvectors of a compact, convex body Definition of belt bodies Solution of the illuminations problem for belt bodies Solution of the Szefalvi-Nagy problem for belt bodies Minumal fixing systems VIII Some research problems Bibliography Author Index Subject Index List of Symbols