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Let $C \in \mathbb{R}^n$, $D > 0$, and $C_1,\ldots,C_{n+1}$ some distinct points with $\|C - C_k\| = D$. Consider the balls $\mathcal{B}(C_k,r_k)$ for some $r_k > 0$. Can we prove that $$\left| \bigcap_{k=1}^{n+1} \partial \mathcal{B}(C_k,r_k) \right| \leq n$$ i.e the intersection has at most $n$ elements?


Later edit:

It might be true even for an intersection of $3$ such spheres ...

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  • $\begingroup$ "it has a unique element"- what is "it"? $\endgroup$
    – whoisit
    Nov 4, 2022 at 12:11
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    $\begingroup$ The intersection $\endgroup$
    – C Marius
    Nov 4, 2022 at 12:11
  • $\begingroup$ and what is $\partial \mathcal{B}(\cdot, \cdot)$? $\endgroup$
    – whoisit
    Nov 4, 2022 at 12:12
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    $\begingroup$ You could first try constructing such an intersection for $n=1,$ $n=2,$ $n=3$ and see how things work in low dimensions. $\endgroup$
    – David K
    Nov 4, 2022 at 12:12
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    $\begingroup$ $\partial \mathcal{B}(C,R)$ is the frontier of the ball. Is the way I designate a sphere $\endgroup$
    – C Marius
    Nov 4, 2022 at 12:19

1 Answer 1

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This paper shows that the intersection of $n$ spheres in $\mathbb{R}^n$ with affinely independent centers has at most $2$ elements. Now only remains to answer the question if $n$ distinct points on a sphere are always affinely independent? The answer is NO!

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