Abstract

Using Minkowski addition of sets, we study linear betweenness in the hyperspace L(X)L(X) of linearly convex nonempty subsets of a normed real vector space XX, as well as in the sub-hyperspace KL(X)KL(X) of compact elements of L(X)L(X). We also study the metric betweenness relation induced by the Hausdorff metric on the latter. While linear betweenness in L(X)L(X) behaves reasonably like linear betweenness at the point level, the analogy is not perfect: linear intervals in XX are honest line segments; this is no longer the case for L(X)L(X), where linear intervals can have exactly two elements. However, when we restrict our focus to KL(X)KL(X), the Rådström extension theorem allows us to view this hyperspace as a linearly convex cone in a normed vector space R(X)\mathcal{R}(X); in particular, all linear intervals are line segments that are contained in the corresponding metric intervals. 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)KL(X) are linearly convex, metric convexity has more nuanced behaviour. For example, the metric intervals in KL(X)KL(X) determined by singletons are all metrically convex if and only if XX is strictly convex. When XX is one-dimensional, R(X)\mathcal{R}(X) is Cartesian 2-space equipped with the max\textit{max} norm and KL(X)KL(X) looks like the half-plane {⟨x,y⟩:x≤y}\{\langle x,y\rangle: x\leq y\}. In particular, all metric intervals – and no closed balls of positive radius – are metrically convex. When XX 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.

Author information

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Keywords

  • Betweenness
  • betweenness axioms
  • convexity
  • vector spaces
  • Minkowski addition
  • metric spaces
  • normed vector spaces
  • strict convexity
  • hyperspaces
  • Hausdorff metric
  • Rådström extension

Mathematics Subject Classification

51K05; 46B20, 52A01, 52A10, 52A21, 52A30, 54A05, 54E35

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Suggested citation

D. Anderson, P. Bankston, A. McCluskey. “Betweenness-Induced Convexity in Hyperspaces of Normed Vector Spaces.” Journal of Convex Analysis 32 (2025), No. 1, 25–60. https://doi.org/10.68381/jca32004

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.