Ball and Spindle Convexity with respect to a Convex Body

Let $C\subset {\mathbb{R}}^n$ 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 $\mathrm{\varnothing }$, or ${\mathbb R}^n$\mathrm{.}The$C$-ballconvexhulloftwopointsiscalleda$C$-spindle\mathrm{.}$K$is$C$-spindleconvexifitcontainsthe$C$-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 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.