Isoperimetry and Integrability of the Sum of Independent Banach-Space Valued Random Variables

We develop a new method to study the tails of a sum of independent mean zero Banach-space valued random variables $(X_i)_{i \leq N}.$ It relies on a new isoperimetric inequality for subsets of a product of probability spaces. In particular, we prove that for $p \geq 1,$ $\bigg\|\sum_{i \leq N} X_i\bigg\|_p \leq \frac{Kp}{1 + \log p}\bigg(\bigg\|\sum_{i \leq N} X_i\bigg\|_1 + \|\max_{i \leq N}\|X_i\|\|_p\bigg),$ where $K$ is a universal constant. Other optimal inequalities for exponential moments are obtained.