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Mat. Zametki, 2001, Volume 70, Issue 5, Pages 679–690 (Mi mz780)  

This article is cited in 6 scientific papers (total in 6 papers)

On a Property of Functions on the Sphere

A. Yu. Volovikov

Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University)

Abstract: According to the Knaster conjecture, for any continuous function $f\colon S^{n-1}\to\mathbb R$ and any $n$-point subset of the sphere $S^{n-1}$, there exists a rotation mapping all the points of this subset to a level surface of the function $f$. In the present paper, this conjecture is proved for the case in which $n=p^2$ for an odd prime $p$ and the points lie on a circle and divide it into equal parts.

DOI: https://doi.org/10.4213/mzm780

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English version:
Mathematical Notes, 2001, 70:5, 616–627

Bibliographic databases:

UDC: 515.142.226
Received: 06.04.1999
Revised: 27.06.2000

Citation: A. Yu. Volovikov, “On a Property of Functions on the Sphere”, Mat. Zametki, 70:5 (2001), 679–690; Math. Notes, 70:5 (2001), 616–627

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. Yu. Volovikov, “Equivariant Maps and Some Problems of the Geometry of Convex Sets”, Proc. Steklov Inst. Math., 239 (2002), 74–87  mathnet  mathscinet  zmath
    2. A. Yu. Volovikov, “Coincidence points of maps of $\mathbb Z_p^n$-spaces”, Izv. Math., 69:5 (2005), 913–962  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. Hinrichs, A, “The Knaster problem: More counterexamples”, Israel Journal of Mathematics, 145 (2005), 311  crossref  mathscinet  zmath  isi  scopus  scopus
    4. R. N. Karasev, “Topological methods in combinatorial geometry”, Russian Math. Surveys, 63:6 (2008), 1031–1078  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. Liu Yu., “On a Property of Functions on the Sphere and its Application”, Nonlinear Anal.-Theory Methods Appl., 73:10 (2010), 3376–3381  crossref  mathscinet  zmath  isi  scopus  scopus
    6. Liu Yu., “Some Mapping Theorems for Continuous Functions Defined on the Sphere”, Nonlinear Anal.-Theory Methods Appl., 75:4 (2012), 1881–1886  crossref  mathscinet  zmath  isi  scopus  scopus
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