The following statement seems clearly true, but I'm having a hard time proving it:
Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. For $r\ge 0$, let $B(c,r)\equiv[c-r,c+r]$. Fix $r_1,\ldots,r_n\in[0,\infty)$, and $c_1,\ldots,c_n\in\mathbb R$. Then, $\mu\left(\bigcap_{i=1}^{n}B(\alpha c_{i},r_{i})\right)\ge\mu\left(\bigcap_{i=1}^{n}B(c_{i},r_{i})\right)$
In words, the statement just says that contracting the 1-dimensional balls' centers towards $0$, by a constant proportion, while leaving the radii unchanged, increases the amount of overlap. This seems straightforward enough: if the centers were contracted all the way to the origin, then the intersection would just be the smallest ball. There can't be more overlap than that.
Can anyone suggest a simple proof, or point me to a relevant theorem?