Excursion into combinatorial geometry

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The book covers a comprehensive exploration of convexity and its various aspects across multiple dimensions. It begins with an in-depth analysis of convex sets, faces, supporting hyperplanes, and the concept of polarity, leading to discussions on the lower semicontinuity of the exponential operator and convex cones. The Farkas Lemma and its generalizations are examined, along with separable systems of convex cones. The focus then shifts to d-convexity in normed spaces, defining d-convex sets and their support properties, and exploring the separability and Helly dimension of these sets. The section on H-convexity introduces the functional md for vector systems, the ?-displacement Theorem, and the continuity properties of H-convex sets, along with applications and connections to d-convexity. The Szökefalvi-Nagy Problem is addressed, detailing the theorem and its generalizations, as well as descriptions of vector systems and compact convex bodies. Borsuk’s partition problem is formulated and surveyed, particularly in relation to bodies of constant width in various spaces. The book also delves into homothetic covering and illumination, discussing key problems and results, including inner illumination of convex bodies. It concludes with a focus on the combinatorial geometry of belt bodies, presenting integral representations, definitions, and solutions to relevant problems. The work is rounded off with an author index and a list