Measure growth in compact semisimple Lie groups and the Kemperman Inverse Problem

Suppose $G$ is a compact semisimple Lie group, $\mu$ is the normalized Haar measure on $G$, and $A,A^2\subseteq G$ are measurable. We show that $$\mu (A^2)\ge \min \{1,2\mu (A)+\eta \mu (A)(1-2\mu (A))\}$$ with the absolute constant $\eta >0$ (independent from the choice of $G$) quantitatively determined. We also show a more general result for connected compact groups without a toric quotient and resolve the Kemperman Inverse Problem from 1964.