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Borsuk--Ulam type theorem for f--neighbors Musin, Oleg


We introduce and study a new class of extensions for the Borsuk--Ulam theorem. Our approach is based on the theory of Voronoi diagrams and Delaunay triangulations. One of our main results is as follows. \begin{thm}\label{corDln1} Let $S^m$ be a unit sphere in $R^{m+1}$ and let $f: S^m \to R^n$ be a continuous map. Then there are points $p$ and $q$ in $S^m$ such that \begin{itemize} \item $\|p-q\|\ge\sqrt{2\cdot\frac{m+2}{m+1}}$\/{\rm;} \item $f(p)$ and $f(q)$ lie on the boundary of a closed metric ball $B$ in $R^n$ whose interior does not meet $f(S^m)$. \end{itemize} \end{thm} Note that $\sqrt{2\cdot\frac{m+2}{m+1}}$ is the diameter of a regular simplex inscribed in $S^m$. Joint paper with Andrey Malyutin.

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