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

Agnieszka Bogdewicz, and Jerzy Grzybowski. "Pairs of convex bodies in a hyperspace over a Minkowski two-dimensional space joined by a unique metric segment." Banach Center Publications 84.1 (2009): 75-88. <http://eudml.org/doc/281750>.

@article{AgnieszkaBogdewicz2009,
abstract = {Let $(ℝ,||·||_)$ be a Minkowski space with a unit ball and let $ϱ_H^\{\}$ be the Hausdorff metric induced by $||·||_\{\}$ 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 $ϱ_H^\{Bⁿ\}$ for the Euclidean unit ball Bⁿ. We extend Schneider’s theorem to the hyperspace $(²,ϱ_H^\{\})$ over any two-dimensional Minkowski space.},
author = {Agnieszka Bogdewicz, Jerzy Grzybowski},
journal = {Banach Center Publications},
keywords = {convex body; Minkowski space; metric midpoint; metric segment},
language = {eng},
number = {1},
pages = {75-88},
title = {Pairs of convex bodies in a hyperspace over a Minkowski two-dimensional space joined by a unique metric segment},
url = {http://eudml.org/doc/281750},
volume = {84},
year = {2009},
}

TY - JOUR
AU - Agnieszka Bogdewicz
AU - Jerzy Grzybowski
TI - Pairs of convex bodies in a hyperspace over a Minkowski two-dimensional space joined by a unique metric segment
JO - Banach Center Publications
PY - 2009
VL - 84
IS - 1
SP - 75
EP - 88
AB - Let $(ℝ,||·||_)$ be a Minkowski space with a unit ball and let $ϱ_H^{}$ be the Hausdorff metric induced by $||·||_{}$ 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 $ϱ_H^{Bⁿ}$ for the Euclidean unit ball Bⁿ. We extend Schneider’s theorem to the hyperspace $(²,ϱ_H^{})$ over any two-dimensional Minkowski space.
LA - eng
KW - convex body; Minkowski space; metric midpoint; metric segment
UR - http://eudml.org/doc/281750
ER -