On a Property of Functions on the Sphere

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.