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

Leighton Pate Barnes, Ayfer Özgür

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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> norm of T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> f for 1 ≤ α ≤ 2 where T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> -norm related quantity can cross zero as a function of α, and show that these two conjectures are essentially equivalent.