bibkey: borsuk1933spheres
authors: Karol Borsuk
year: 1933
title: "Drei Sätze über die n-dimensionale euklidische Sphäre"
doi: 10.4064/fm-20-1-177-190
url: "https://matwbn.icm.edu.pl/ksiazki/fm/fm20/fm20117.pdf"
claim: "The paper proves three theorems on the n-dimensional sphere; the second states that every continuous map of the n-sphere into n-dimensional Euclidean space sends some pair of antipodal points to the same point."
strata_touched:
  - D5/S3/Geometry/Lorentz/ChiralEventParityMinimality
license: citation-only
triage: anchor

Borsuk, Drei Sätze über die n-dimensionale euklidische Sphäre

K. Borsuk, Fundamenta Mathematicae 20 (1933) 177–190, DOI 10.4064/fm-20-1-177-190. The quotation is from the scan in the ICM digital library.

$S_n$ is the boundary of the unit ball in ${\mathbb{R}}^{n+1}$, and $p^{\ast }$ is the point antipodal to $p$. The paper states at the outset (p. 178):

Satz II. Ist $f\in R^{n^{S_n}}$ (d. h. bildet $f$ die Sphäre $S_n$ auf einen Teil von $R^n$ ab), so gibt es einen derartigen Punkt $p\in S_n$, dass $f(p)=f(p^{\ast })$ ist.

A footnote records that the theorem was conjectured by S. Ulam. For an odd map, $f(p^{\ast })=-f(p)$, so Satz II gives a point with $f(p)=0$.

Verified locator

Source: https://matwbn.icm.edu.pl/ksiazki/fm/fm20/fm20117.pdf

• DOI: https://doi.org/10.4064/fm-20-1-177-190 (title, author, journal, volume and pages from Crossref).
• URL: https://matwbn.icm.edu.pl/ksiazki/fm/fm20/fm20117.pdf (the scanned article in the ICM digital library). Locations: the definitions of $S_n$ and $p^{\ast }$, p. 177; Satz II and footnote 7, p. 178.