On a discrete version of the antipodal theorem

• Autorzy: Krzysztof Oleszkiewicz
• Języki publikacji: EN

Abstrakty

EN

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]).

Kategorie tematyczne

• 54H25: Fixed-point and coincidence theorems
• 05B25: Finite geometries
• 46B09: Probabilistic methods in Banach space theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1996
• Tom: 151
• Numer: 2
• Strony: 189-194

Daty

wydano

1996

otrzymano

1996-04-03

poprawiono

1996-06-24

Twórcy

autor

Krzysztof Oleszkiewicz

• Department of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland

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).