In 1989 R. Arnold proved that for every pair (A,B) of compact convex subsets of ℝ there is an Euclidean isometry optimal with respect to L₂ metric and if f₀ is such an isometry, then the Steiner points of f₀(A) and B coincide. In the present paper we solve related problems for metrics topologically equivalent to the Hausdorff metric, in particular for metrics for all p ≥ 2 and the symmetric difference metric.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc84-0-10,
author = {Irmina Herburt and Maria Moszy\'nska and Dorette Pronk},
title = {Fractal star bodies},
journal = {Banach Center Publications},
volume = {86},
year = {2009},
pages = {149-171},
zbl = {1184.28005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc84-0-10}
}
Irmina Herburt; Maria Moszyńska; Dorette Pronk. Fractal star bodies. Banach Center Publications, Tome 86 (2009) pp. 149-171. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc84-0-10/