Local Optimality of Dictator Functions with Applications to Courtade--Kumar Conjecture

Given a convex function $\Phi:[0,1]\to\mathbb{R}$, the $\Phi$-stability of a Boolean function $f$ is $\mathbb{E}[\Phi(T_{\rho}f(\mathbf{X}))]$, where $\mathbf{X}$ is a random vector uniformly distributed on the discrete cube $\{\pm1\}^{n}$ and $T_{\rho}$ is the Bonami-Beckner operator. In this paper, we prove that dictator functions are locally optimal in maximizing the $\Phi$-stability of $f$ over all balanced Boolean functions. Combining this result with our previous bound in \cite{yu2023phi}, we provide a new bound for the Courtade--Kumar conjecture which is expressed in the form of finite-dimensional program. By evaluating this new bound, we numerically verify that the Courtade--Kumar conjecture is true for all $\rho\in[0,0.92]$. Our proofs are based on the majorization of noise operators and hypercontractivity inequalities.