MHB Show Metric Proves All Points Inside Circle Have Same Center

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The discussion centers on proving that all points within a circle share the same center, using the metric $d_p(n+m)=|n-m|_p$. Participants clarify that the correct notation for the metric should be $d_p(m,n)=|n-m|_p$, which refers to the $p$-adic metric on the rational numbers. There is a request for a definition of a circle's center and clarification on the variable $p$. A link to a relevant resource is provided for further exploration of the topic. The conversation emphasizes the importance of precise definitions and notation in mathematical discussions.
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Show that all point inside of an circle is his center. Consider the metric $d_p(n+m)=|n-m|_p.$

Hello MHB :). Any hint for the problem?, thanks!.
 
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Hi, Julio.

Julio said:
Show that all point inside of an circle is his center. Consider the metric $d_p(n+m)=|n-m|_p.$
Could you give the definition of a circle center? Also, a metric is a function of two arguments, while $d_p(n+m)$ has one argument. Finally, what is $p$?
 
Julio said:
Show that all point inside of an circle is his center. Consider the metric $d_p(n+m)=|n-m|_p.$
I suppose you mean $d_p(m,n)=|n-m|_p$ where $d_p$ is the $p$-adic metric on $\mathbb{Q},$ and disc instead of circle. If so, have a look https://www.colby.edu/math/faculty/Faculty_files/hollydir/Holly01.pdf.
 
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