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Inhaltsangabe
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.
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.
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