The case of equality in BK

We characterize the pairs of increasing events $A,B$ for which there is equality in the BK inequality. Namely, we show that $P(A\circ B)=P(A)P(B)$ if and only if all the configurations in $A\times B$ admit disjoint witnesses for $A$ and $B$. We discuss the strengthened BK inequality, and we provide a new simplified proof of this inequality.