The classical theorem of Borsuk and Ulam [2] says that for any continuous mapping there exists a point such that f(-x) = f(x). In this note a discrete version of the antipodal theorem is proved in which is replaced by the set of vertices of a high-dimensional cube equipped with Hamming’s metric. In place of equality we obtain some optimal estimates of which were previously known (as far as the author knows) only for f linear (cf. [1]).
@article{bwmeta1.element.bwnjournal-article-fmv151i2p189bwm,
author = {Krzysztof Oleszkiewicz},
title = {On a discrete version of the antipodal theorem},
journal = {Fundamenta Mathematicae},
volume = {149},
year = {1996},
pages = {189-194},
zbl = {0879.54049},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv151i2p189bwm}
}
Oleszkiewicz, Krzysztof. On a discrete version of the antipodal theorem. Fundamenta Mathematicae, Tome 149 (1996) pp. 189-194. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv151i2p189bwm/
[00000] [1] I. Bárány and V. S. Grinberg, On some combinatorial questions in finite-dimensional spaces, Linear Algebra Appl. 41 (1981), 1-9. | Zbl 0467.90079
[00001] [2] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966, Theorem 5.8.9.
[00002] [3] C.-T. Yang, On a theorem of Borsuk-Ulam, Ann. of Math. 60 (1954), 262-282, Theorem 1, (2.7), (3.1). | Zbl 0057.39104