Ball and spindle convexity with respect to a convex body

Lángi, Zsolt and Naszódi, Márton and Talata, István (2012) Ball and spindle convexity with respect to a convex body. Aequationes Mathematicae, 85 (1-2). pp. 41-67. ISSN 0001-9054 (print version), 1420-8903 (electronic version)

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Abstract

Let C⊂R^n be a convex body. We introduce two notions of convexity associated to C. A set K is C-ball convex if it is the intersection of translates of C, or it is either ∅ , or R^n . The C-ball convex hull of two points is called a C-spindle. K is C-spindle convex if it contains the C-spindle of any pair of its points. We investigate how some fundamental properties of conventional convex sets can be adapted to C-spindle convex and C-ball convex sets. We study separation properties and Carathéodory numbers of these two convexity structures. We investigate the basic properties of arc-distance, a quantity defined by a centrally symmetric planar disc C, which is the length of an arc of a translate of C, measured in the C-norm that connects two points. Then we characterize those n-dimensional convex bodies C for which every C-ball convex set is the C-ball convex hull of finitely many points. Finally, we obtain a stability result concerning covering numbers of some C-ball convex sets, and diametrically maximal sets in n-dimensional Minkowski spaces.

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