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

Tom 84 / 2009

Agnieszka Bogdewicz, Jerzy Grzybowski Banach Center Publications 84 (2009), 75-88 MSC: Primary 52A10, 52A21; Secondary 52A99. DOI: 10.4064/bc84-0-5

Streszczenie

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.

Autorzy

  • Agnieszka BogdewiczFaculty of Mathematics and Information Science
    Warsaw University of Technology
    Pl. Politechniki 1
    00-661 Warszawa, Poland
    e-mail
  • Jerzy GrzybowskiFaculty of Mathematics and Computer Science
    A. Mickiewicz University
    Umultowska 87
    61-614 Poznań, Poland
    e-mail

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