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Let (R, ‖ · ‖B) be a Minkowski space with a unit ball B and let %H be the Hausdorff metric induced by ‖ · ‖B in the hyperspace K of convex bodies (nonempty, compact, convex subsets of R). R. Schneider [3] characterized pairs of elements of K which can be joined by unique metric segments with respect to % n H for the Euclidean unit ball B . We extend Schneider’s theorem to the hyperspace (K, %H) over any two-dimensional Minkowski space. 1. Preliminaries and introduction. A Minkowski space (see [5]) is a finite dimensional normed linear space (R, ‖ · ‖). Let B be the unit ball determined by the norm ‖ · ‖: B := {x ∈ R | ‖x‖ ≤ 1}. Then B is a convex body symmetric at 0 with nonempty interior. Conversely, every convex body A symmetric at 0 with nonempty interior determines a norm, ‖·‖A, usually referred to as the Minkowski functional: ‖x‖A := inf{t ∈ R+ | x ∈ tA} (see [5], p. 17). 2000 Mathematics Subject Classification: Primary 52A10, 52A21; Secondary 52A99.

Pairs of convex bodies in a hyperspace over a Minkowski two-dimensional space joined by a unique metric segment