Geodesic
On a Property of Functions on the Sphere
Matematičeskie zametki, Tome 70 (2001) no. 5, pp. 679-690

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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. 
@article{MZM_2001_70_5_a3,
     author = {A. Yu. Volovikov},
     title = {On a {Property} of {Functions} on the {Sphere}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {679--690},
     publisher = {mathdoc},
     volume = {70},
     number = {5},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2001_70_5_a3/}
}
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A. Yu. Volovikov. On a Property of Functions on the Sphere. Matematičeskie zametki, Tome 70 (2001) no. 5, pp. 679-690. http://geodesic.mathdoc.fr/item/MZM_2001_70_5_a3/