# On a discrete version of the antipodal theorem

Fundamenta Mathematicae (1996)

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

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topOleszkiewicz, 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

top- [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] 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). Zbl0057.39104

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