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

Agnieszka Bogdewicz; Jerzy Grzybowski — 2009

Banach Center Publications

Let $(ℝ, | | \cdot | |_{)}$ be a Minkowski space with a unit ball and let $ϱ_{H}^{}$ be the Hausdorff metric induced by ${| | \cdot | |}_{}$ in the hyperspace of convex bodies (nonempty, compact, convex subsets of ℝ). R. Schneider [RSP] characterized pairs of elements of which can be joined by unique metric segments with respect to $ϱ_{H}^{B ⁿ}$ for the Euclidean unit ball Bⁿ. We extend Schneider’s theorem to the hyperspace $(², ϱ_{H}^{})$ over any two-dimensional Minkowski space.

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