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

04/02/2020

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by   Leighton Pate Barnes, et al.

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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^α norm of T_pf for 1≤α≤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^α-norm related quantity can cross zero as a function of α, and show that these two conjectures are essentially equivalent.