On a discrete version of the antipodal theorem

Krzysztof Oleszkiewicz

Fundamenta Mathematicae (1996)

  • Volume: 151, Issue: 2, page 189-194
  • ISSN: 0016-2736

Abstract

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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 i n f x | | f ( x ) - f ( - x ) | | which were previously known (as far as the author knows) only for f linear (cf. [1]).

How to cite

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Oleszkiewicz, Krzysztof. "On a discrete version of the antipodal theorem." Fundamenta Mathematicae 151.2 (1996): 189-194. <http://eudml.org/doc/212190>.

@article{Oleszkiewicz1996,
abstract = {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]).},
author = {Oleszkiewicz, Krzysztof},
journal = {Fundamenta Mathematicae},
keywords = {antipodal theorem},
language = {eng},
number = {2},
pages = {189-194},
title = {On a discrete version of the antipodal theorem},
url = {http://eudml.org/doc/212190},
volume = {151},
year = {1996},
}

TY - JOUR
AU - Oleszkiewicz, Krzysztof
TI - On a discrete version of the antipodal theorem
JO - Fundamenta Mathematicae
PY - 1996
VL - 151
IS - 2
SP - 189
EP - 194
AB - 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]).
LA - eng
KW - antipodal theorem
UR - http://eudml.org/doc/212190
ER -

References

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  1. [1] I. Bárány and V. S. Grinberg, On some combinatorial questions in finite-dimensional spaces, Linear Algebra Appl. 41 (1981), 1-9. Zbl0467.90079
  2. [2] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966, Theorem 5.8.9. 
  3. [3] C.-T. Yang, On a theorem of Borsuk-Ulam, Ann. of Math. 60 (1954), 262-282, Theorem 1, (2.7), (3.1). Zbl0057.39104

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