A new asymmetric correlation inequality for Gaussian measure

Overview & Synopsis

The Khatri-\v{S}id\'{a}k lemma says that for any Gaussian measure $\mu$ over $\mathbb{R}^n$ , given a convex set $K$ and a slab $L$, both symmetric about the origin, one has $\mu(K \cap L) \geq \mu(K)\mu(L)$. We state and prove a new asymmetric version of the Khatri-\v{S}id\'{a}k lemma when $K$ is a symmetric convex body and $L$ is a slab (not necessarily symmetric about the barycenter of $K$). Our result also extends that of Szarek and Werner (1999), in a special case.