Can anyone prove this statement? It seems true, but I'm finding it tricky to give a concise proof.

Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. Define $B(c,r)\equiv[c-r,c+r]$, where $[\cdot, \cdot]$ denotes an interval. For $i=1,\ldots,n$, fix $r_1,\ldots,r_n\in[0,\infty)$, and $c_1,\ldots,c_n\in\mathbb R$. Let $\bar c, \bar r, \tilde c$, and $\tilde r$ satisfy $B(\bar c, \bar r)=\bigcap_{i=1}^{n}B(c_{i},r_{i})$ and $B(\tilde c, \tilde r)=\bigcap_{i=1}^{n}B(c_{i},\alpha r_{i})$. Then, $\tilde r \le \alpha r$.