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Borsuk--Ulam type theorem for f--neighbors Musin, Oleg
Description
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.
Item Metadata
Title |
Borsuk--Ulam type theorem for f--neighbors
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2019-10-10T09:46
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Description |
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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Extent |
43.0 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of Texas Rio Grande Valley
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Series | |
Date Available |
2020-04-08
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0389768
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International