The Courtade-Kumar Most Informative Boolean Function Conjecture and a Symmetrized Li-M\'edard Conjecture are Equivalent

Leighton Pate Barnes; Ayfer Özgür

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arXiv · 2004.01277

The Courtade-Kumar Most Informative Boolean Function Conjecture and a Symmetrized Li-M\'edard Conjecture are Equivalent

Leighton Pate Barnes·Ayfer Özgür

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Abstract

We consider the Courtade-Kumar most informative Boolean function conjecture for balanced functions, as well as a conjecture by Li and M\'edard that dictatorship functions also maximize the $L^\alpha$ norm of $T_pf$ for $1\leq\alpha\leq2$ 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.

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Leighton Pate Barnes, Ayfer Özgür. 2020-04-02. The Courtade-Kumar Most Informative Boolean Function Conjecture and a Symmetrized Li-M\'edard Conjecture are Equivalent. https://arxiv.org/abs/2004.01277

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