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We prove a quantitative form of the celebrated Ball's theorem on cube slicingin $\mathbb{R}^n$ and obtain, as a consequence, equality cases in themin-entropy power inequality. Independently, we also give a quantitative formof Khintchine's inequality in the special case $p=1$.

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