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Prove that in $R^n$ with the euclidean norm there are no two identical open balls $B_r(q)$ and $B_r(p)$ such that $p\ne q$.

I believe to do this I need to show that if you have balls around two different points of the same radii, that this implies there is some point in one but not the other. However, I can't come up with how to come up with this point in the general case.

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    Interestingly enough, this is false in a spectacular way for some metric spaces. In an ultrametric space such as the $p$-adic numbers, one actually has $B_r(p) = B_r(q)$ whenever $p \in B_r(q)$; in other words, every point in a ball is a centre of the ball!2012-12-23

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Without loss of generality $p=(-1,0, \dots, 0)$ and $q=(1,0,\dots,0)$. Then $a=(-(1+r),0,\dots, 0)$ is in the closed ball with centre $p$ and radius $r$. But the distance between $q$ and $a$ is $2+r\gt r$. So $a$ is in the closed ball with centre $p$ and radius $r$, but is not in the closed ball with centre $q$ and radius $r$.

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    Yes, a basic property of Euclidean norm is that it is invariant under translation, rotation.2012-12-23
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Just take the straight line connecting the centers $ p $ and $ q $. If you proceed from $ p $ to $ q $ along this straight line, then after some point, you will leave $ {\mathbb{B}_{r}}(p) $ but either enter or stay within $ {\mathbb{B}_{r}}(q) $.