Betweenness-Induced Convexity in Hyperspaces of Normed Vector Spaces

Using Minkowski addition of sets, we study linear betweenness in the hyperspace $L(X)$ of linearly convex nonempty subsets of a normed real vector space $X$, as well as in the sub-hyperspace $KL(X)$ of compact elements of $L(X)$. We also study the metric betweenness relation induced by the Hausdorff metric on the latter. While linear betweenness in $L(X)$ behaves reasonably like linear betweenness at the point level, the analogy is not perfect: linear intervals in $X$ are honest line segments; this is no longer the case for $L(X)$, where linear intervals can have exactly two elements. However, when we restrict our focus to $KL(X)$, the R\r{a}dstr\"{o}m extension theorem allows us to view this hyperspace as a linearly convex cone in a normed vector space $\mathcal{R}(X)$; in particular, all linear intervals are line segments that are contained in the corresponding metric intervals.
[1mm] We are especially interested in the notions of convexity induced by these two kinds of betweenness relation. While all closed balls and metric intervals in $KL(X)$ are linearly convex, metric convexity has more nuanced behaviour. For example, the metric intervals in $KL(X)$ determined by singletons are all metrically convex if and only if $X$ is strictly convex. When $X$ is one-dimensional, $\mathcal{R}(X)$ is Cartesian 2-space equipped with the max norm and $KL(X)$ looks like the half-plane $\{\langle x,y\rangle: x\leq y\}$. In particular, all metric intervals -- and no closed balls of positive radius -- are metrically convex. When $X$ is multi-dimensional, though, while it is still the case that closed balls are metrically nonconvex, it is now always possible to find a metrically nonconvex metric interval that is determined by a singleton and a line segment.