Journal of Convex Analysis 32 (2025), No. 1, 025--060
Copyright Heldermann Verlag 2025
Betweenness-Induced Convexity in Hyperspaces of Normed Vector Spaces
Daron Anderson
School of Computer Science, Trinity College, Dublin, Ireland
daronanderson@live.ie
Paul Bankston
Dept. of Mathematical and Statistical Sciences, Marquette University, Milwaukee, U.S.A.
paul.bankston@marquette.edu
Aisling McCluskey
School of Mathematical and Statistical Sciences, University of Galway, Ireland
aisling.mccluskey@universityofgalway.ie
[Abstract-pdf]
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 \textit{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.
Keywords: Betweenness, betweenness axioms, convexity, vector spaces, Minkowski addition, metric spaces, normed vector spaces, strict convexity, hyperspaces, Hausdorff metric, Radstroem extension.
MSC: 51K05; 46B20, 52A01, 52A10, 52A21, 52A30, 54A05, 54E35.
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