Ball and spindle convexity with respect to a convex body

Let C ⊂ Rn be a convex body. We introduce two notions of convexity associated to C. A set K is C-ball convex if it is the inter- section of translates of C, or it is either ∅, or Rn. The C-ball convex hull of two points is called a C-spindle. K is C-spindle convex if it con- tains 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 proper- ties and Carath´eodory 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.