Pairs of convex bodies in a hyperspace over a Minkowski two-dimensional space joined by a unique metric segment
Volume 84 / 2009
Banach Center Publications 84 (2009), 75-88
MSC: Primary 52A10, 52A21; Secondary 52A99.
DOI: 10.4064/bc84-0-5
Abstract
Let $({\mathbb R}, \| \cdot \|_\mathbb{B})$ be a Minkowski space with a unit ball $\mathbb{B}$ and let $\varrho_H^{\mathbb{B}}$ be the Hausdorff metric induced by $\|\cdot \|_{\mathbb{B}}$ in the hyperspace ${{\cal K}}$ of convex bodies (nonempty, compact, convex subsets of ${\mathbb R}$). R. Schneider \cite {RSP} characterized pairs of elements of ${{\cal K}}$ which can be joined by unique metric segments with respect to $\varrho_H^{B^{n}}$ for the Euclidean unit ball $B^{n}$. We extend Schneider's theorem to the hyperspace $(\mathcal{K}^{2},\varrho_H^{\mathbb{B}})$ over any two-dimensional Minkowski space.
