Suppose $G$ is a compact semisimple Lie group, $μ$ is the normalized Haar measure on $G$, and $A, A^2 \subseteq G$ are measurable. We show that $$μ(A^2)\geq \min\{1, 2μ(A)+ημ(A)(1-2μ(A))\}$$ with the absolute constant $η>0$ (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.