Ball and spindle convexity with respect to a convex body

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.