Quantitative form of Balls Cube slicing in $mathbb{R}^n$ and equality cases in the min-entropy power inequality

We prove a quantitative form of the celebrated Balls theorem on cube slicing in $mathbb{R}^n$ and obtain, as a consequence, equality cases in the min-entropy power inequality. Independently, we also give a quantitative form of Khintchines inequality in the special case $p=1$.