We examine minimality in asymmetry classes of convex compact sets with respect to inclusion. We prove that each class has a minimal element. Moreover, we show there is a connection between asymmetry classes and the Rådström-Hörmander lattice. This is used to present an alternative solution to the problem of minimality posed by G. Ewald and G. C. Shephard in [4].
@article{bwmeta1.element.bwnjournal-article-smv124i2p149bwm,
author = {Micha\l\ Wiernowolski},
title = {Minimality in asymmetry classes},
journal = {Studia Mathematica},
volume = {122},
year = {1997},
pages = {149-154},
zbl = {0883.52001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv124i2p149bwm}
}
Wiernowolski, Michał. Minimality in asymmetry classes. Studia Mathematica, Tome 122 (1997) pp. 149-154. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv124i2p149bwm/
[00000] [1] J. Grzybowski, Minimal pairs of convex compact sets, Arch. Math. (Basel) 63 (1994), 173-181. | Zbl 0804.52002
[00001] [2] D. Pallaschke, S. Scholtes and R. Urbański, On minimal pairs of convex compact sets, Bull. Polish Acad. Sci. Math. 39 (1991), 1-5. | Zbl 0759.52003
[00002] [3] R. Schneider, On asymmetry classes of convex bodies, Mathematika 21 (1974), 12-18. | Zbl 0288.52003
[00003] [4] G. C. Shephard and G. Ewald, Normed vector spaces consisting of classes of convex sets, Math. Z. 91 (1966), 1-19. | Zbl 0141.39003
[00004] [5] R. Urbański, A generalization of the Minkowski-Rådström-Hörmander Theorem, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), 709-715. | Zbl 0336.46009