$\psi _{\alpha }$-estimates for marginals of log-concave probability measures

Abstract:

We show that a random marginal $\pi _F(\mu )$ of an isotropic log-concave probability measure $\mu$ on $\mathbb R^n$ exhibits better $\psi _{\alpha }$-behavior. For a natural variant $\psi _{\alpha }^{\prime }$ of the standard $\psi _{\alpha }$-norm we show the following:

1. [(i)] If $k\leq \sqrt {n}$, then for a random $F\in G_{n,k}$ we have that $\pi _F(\mu )$ is a $\psi _2^{\prime }$-measure. We complement this result by showing that a random $\pi _F(\mu )$ is, at the same time, super-Gaussian.

2. [(ii)] If $k=n^{\delta }$, $\frac {1}{2}<\delta <1$, then for a random $F\in G_{n,k}$ we have that $\pi _F(\mu )$ is a $\psi _{\alpha (\delta )}^{\prime }$-measure, where $\alpha (\delta )=\frac {2\delta }{3\delta -1}$.