The Courtade-Kumar Most Informative Boolean Function Conjecture and a Symmetrized Li-Médard Conjecture are Equivalent

Abstract

We consider the Courtade-Kumar most informative Boolean function conjecture for balanced functions, as well as a conjecture by Li and Médard that dictatorship functions also maximize the $L^{\alpha }$ norm of $T_{p}f$ for $1\le \alpha \le 2$ where $T_p$ is the noise operator and $f$ is a balanced Boolean function. By using a result due to Laguerre from the 1880's, we are able to bound how many times an $L^{\alpha }$-norm related quantity can cross zero as a function of $\alpha$, and show that these two conjectures are essentially equivalent.