Prove that if the intersection of $n+1$ distinct spheres is non-empty then it has at most $n$ elements in $\mathbb{R}^n$

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 ...

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

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!