The existence of a minimal element in every equivalence class of pairs of bounded closed convex sets in a reflexive locally convex topological vector space is proved. An example of a non-reflexive Banach space with an equivalence class containing no minimal element is presented.
@article{bwmeta1.element.bwnjournal-article-smv126i1p95bwm,
author = {J. Grzybowski and R. Urba\'nski},
title = {Minimal pairs of bounded closed convex sets},
journal = {Studia Mathematica},
volume = {122},
year = {1997},
pages = {95-99},
zbl = {0896.52002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv126i1p95bwm}
}
Grzybowski, J.; Urbański, R. Minimal pairs of bounded closed convex sets. Studia Mathematica, Tome 122 (1997) pp. 95-99. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv126i1p95bwm/
[00000] [1] V. F. Dem'yanov and A. M. Rubinov, Quasidifferential Calculus, Optimization Software Inc., New York, 1986.
[00001] [2] J. Grzybowski, Minimal pairs of compact convex sets, Arch. Math. (Basel) 63 (1994), 173-181. | Zbl 0804.52002
[00002] [3] D. Pallaschke, S. Scholtes and R. Urbański, On minimal pairs of compact convex sets, Bull. Polish Acad. Sci. Math. 39 (1991), 1-5. | Zbl 0759.52003
[00003] [4] S. Scholtes, Minimal pairs of convex bodies in two dimensions, Mathematika 39 (1992), 267-273. | Zbl 0759.52004
[00004] [5] R. Urbański, A generalization of the Minkowski-Rå dström-Hörmander theorem, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), 709-715. | Zbl 0336.46009
[00005] [6] M. Wiernowolski, On amount of minimal pairs, Funct. Approx. Comment. Math. 23 (1994), 35-39. | Zbl 0838.52005