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

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

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

Thank you very much for your help.