Titlebook: Join Geometries; A Theory of Convex S Walter Prenowitz,James Jantosciak Textbook 1979 Springer-Verlag New York Inc. 1979 Equivalence.Factor

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0172-6056 porary mathematics. The postulational basis of the subject will be radically revised in order to construct a broad-scale and conceptually unified treatment. The familiar figures of classical geometry-points, segments, lines, planes, triangles, circles, and so on-stem from problems in the physical wo

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Extremal Structure of Convex Sets: Components and Faces, .. The faces of . play a major role in the study of its structure and can be characterized as sections of . by linear spaces which do not “cut” . and are said to be . to it. An application of the theory is made to the characterization of the components and faces of a polytope.

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Cones and Hypercones,eme ray of a convex cone, as an analogue of an extreme point of an arbitrary convex set, is clarified and examined closely. Then the study of cones concludes with the determination of conditions for a polyhedral cone to be generated by its extreme rays.

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The Abstract Theory of Join Operations, their elementary consequences deduced. The notion of convex set is then defined in terms of join and quickly takes on a central role in the theory. With each convex set . there are associated two important subsidiary sets which are themselves convex: the . of . and the . of .. These ideas receive a major share of our attention.

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The Operation of Extension,ar order of points and two categories of convex set referred to as . and .. New results are obtained on familiar ideas: Theorem 4.28—any join of points is an open convex set; Theorem 4.30—the interior of a polytope . is the join of the points of any finite set of generators of .; and Theorem 4.31 − I(.) = I(.)I(.), provided I(.), I(.) ≠ Ø.

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