A Counterexample to a Directed KKL Inequality - Dubroff, Quentin; Nadimpalli, Shivam; Narayanan, Bhargav (The Electronic Journal of Combinatorics, 2024)
We show that the natural directed analogues of the KKL theorem and the Eldan– Gross inequality from the analysis of Boolean functions fail to hold. This is in contrast to several other isoperimetric inequalities on the Boolean hypercube (such as the Poincare inequality, Margulis’s inequality and Talagrand’s inequalit) for which directed strengthenings have recently been established.