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Abstrakty
The classical theorem of Borsuk and Ulam [2] says that for any continuous mapping $f: S^k → ℝ^k$ there exists a point $x ∈ S^k$ such that f(-x) = f(x). In this note a discrete version of the antipodal theorem is proved in which $S^k$ 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 $inf_x ||f(x)-f(-x)||$ which were previously known (as far as the author knows) only for f linear (cf. [1]).
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
189-194
Opis fizyczny
Daty
wydano
1996
otrzymano
1996-04-03
poprawiono
1996-06-24
Twórcy
autor
- Department of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland, koles@mimuw.edu.pl
Bibliografia
- [1] I. Bárány and V. S. Grinberg, On some combinatorial questions in finite-dimensional spaces, Linear Algebra Appl. 41 (1981), 1-9.
- [2] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966, Theorem 5.8.9.
- [3] C.-T. Yang, On a theorem of Borsuk-Ulam, Ann. of Math. 60 (1954), 262-282, Theorem 1, (2.7), (3.1).
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-fmv151i2p189bwm