Show Metric Proves All Points Inside Circle Have Same Center

[Julio1]

Show that all point inside of an circle is his center. Consider the metric $d_p(n+m)=|n-m|_p.$

Spoiler

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

 

[Evgeny.Makarov]

Hi, Julio.

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 $p$?

 

[Fernando Revilla]

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.