Knaster's problem on continuous mappings of a sphere into a Euclidean space

A survey of known results and additional new ones on Knaster's problem: on the standard sphere $S^{n-1}\subset R^n$ find configurations of points $A_1,\dots,A_k$, such that for any continuous map $f\colon S^{n-1}\to R^m$ one can find a rotation $a$ of the sphere $S^{n-1}$ such that $f(a(A_1))=\dotsb=f(a(A_k))$ and some problems closely connected with it. We study the connection of Knaster's problem with equivariant mappings, with Dvoretsky's theorem on the existence of an almost spherical section of a multidimensional convex body, and we also study the set $\{a\in SO(n)\mid f(a(A_1))=\dotsb=f(a(A_k))\}$ of solutions of Knaster's problem for a fixed configuration of points $A_1,\dots,A_k\in S^{n-1}$ and a map $f\colon S^{n-1}\to R^m$ in general position. Unsolved problems are posed.