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I have this problem to solve:

Given N spheres with centers C : {$c_1, ..., c_n| c_i ∈ R^M$} and radiuses R : {$r_1, ..., r_n| r_i ∈ R$}

I'm searching for a sufficient condition which states that exists a region that is part of each sphere.

For N = 2 the condition is simple: if the euclidean distance d($c_1, c_2$) < $r_1 + r_2$ then the region exists

For N > 2 I have difficult both finding a sufficient condition myself or finding it online

Can someone help me?

NooneBug
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  • Welcome to MSE. In order for MathJax commands to be effective, they must be surrounded by $ signs. For example, $x^2$ shows up as $x^2$. – saulspatz Apr 09 '21 at 14:13
  • Thanks, I didn't knew the syntax – NooneBug Apr 09 '21 at 14:15
  • By Helly's theorem, in the plane it's enough to do the $N=3$ case, and in $\mathbb{R}^n$ it's enough to do the $N=n+1$ case. Even the case of three disks in the plane doesn't seem to be easy. I found this paper on computing the area of the overlap. It might have something useful in it. – saulspatz Apr 09 '21 at 14:46

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