In complex analysis the most important domain is the unit disc. In fact all domains (at least simply connected and bounded) are biholomorphic, i.e. analytically equivalent, to the disc.
In higher dimension, the natural analogue is the unit ball. But in higher dimension, the general domain is not biholomorphic to the ball. A basic question is then how well a general domain can be approximated by the ball. If we have a ball $B_r$ of radius $r<1$ contained in the unit ball $B_1$, then a domain $U$ with $B_r\subset U\subset B_1$ is said to be squeezed between the two balls. The larger we can choose $r,$ the closer the domain $U$ is to the ball. I will talk about this topic.
