Let $\displaystyle B_i(a_i,r_i)$ a set of m balls in the euclidean space $\displaystyle R^n$, with
center $\displaystyle a_i$ and radius $\displaystyle r_i$. Let us suppose that they have non-empty

Now consider new centers $\displaystyle b_i$ such that $\displaystyle |b_i-b_j|<|a_i-a_j|$. Then
┐how can I prove that the balls $\displaystyle B_i(b_i,r_i)$ have also non-empty

I have tried by taking the same baricentric coordiantes respect to the centers $\displaystyle a_i$ and later with centers $\displaystyle b_i$. But this doesn't work.

Thank you very much for your help.