A Counterexample to a Directed KKL Inequality

TL;DR

This paper demonstrates that certain directed analogues of well-known inequalities in Boolean function analysis do not hold, highlighting limitations in extending these inequalities to directed settings.

Contribution

It provides a counterexample showing the failure of directed KKL and Eldan-Gross inequalities, contrasting with other isoperimetric inequalities that do extend.

Findings

Directed KKL inequality fails to hold.

Directed Eldan-Gross inequality fails to hold.

Contrasts with other directed isoperimetric inequalities.

Abstract

We show that the natural directed analogues of the KKL theorem [KKL88] and the Eldan--Gross inequality [EG20] 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 [Mar74] and Talagrand's inequality [Tal93]) for which directed strengthenings have recently been established.