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I know that the What is the intersection of two balls in $\mathbb{R}^2$ is either 0,1 or 2 points (since a ball in $\mathbb{R}^2$ is just a circle I can drew this).

What is the intersection of two balls in $\mathbb{R}^3$ ? is it at max 3 points ?

I would also like to know about the more general case of $n>1$

Can someone please help ?

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Two spheres will intersect in a circle. Possibly with zero or imaginary radius. – J. M. Apr 28 '12 at 9:50
A ball is often taken to mean the interior of a sphere, perhaps including the edge, which would make their intersections a lens – Henry Apr 28 '12 at 9:59
up vote 2 down vote accepted

No it will be a circle or a point (or empty), see this wikimedia image:

enter image description here

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It is also possible for the two spheres to not intersect, if e.g. the spheres are concentric or very far apart. – J. M. Apr 28 '12 at 9:57
Of course, I added that part. – Listing Apr 28 '12 at 9:58
Thank you both, the picture really help me see that! – Belgi Apr 28 '12 at 10:08

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