Let $m, n \in \mathbb{N}$. Let $r_1, \ldots, r_m$ be given positive real numbers; for each $i=1, \ldots, m$, let $(x_{i1}, \ldots, x_{in})$ be $m$ given points in $\mathbb{R}^n$, and let $$S_i \colon= \left\{ \ (x_1, \ldots, x_n) \in \mathbb{R}^n \ \colon \sqrt{\sum_{j=1}^n (x_j - x_{ij})^2} < r_i \ \right\}.$$
Finally, let $$S \colon= \bigcap_{i=1}^m S_i.$$
Then
(1) What is (are ) the necessary and sufficient condition(s) --- on the $x_{ij}$, $y_{ij}$, and $r_i$ --- for the intersection set $S$ to be non-empty?