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Let $C\subset {\mathbb R}^n$ be a convex body. We introduce two notions ofconvexity associated to C. A set $K$ is $C$-ball convex if it is theintersection of translates of $C$, or it is either $\emptyset$, or ${\mathbbR}^n$. 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. Weinvestigate how some fundamental properties of conventional convex sets can beadapted to $C$-spindle convex and $C$-ball convex sets. We study separationproperties and Carath\'eodory numbers of these two convexity structures. Weinvestigate the basic properties of arc-distance, a quantity defined by acentrally symmetric planar disc $C$, which is the length of an arc of atranslate of $C$, measured in the $C$-norm, that connects two points. Then wecharacterize those $n$-dimensional convex bodies $C$ for which every $C$-ballconvex set is the $C$-ball convex hull of finitely many points. Finally, weobtain a stability result concerning covering numbers of some $C$-ball convexsets, and diametrically maximal sets in $n$-dimensional Minkowski spaces.

Ball and Spindle Convexity with respect to a Convex Body