The Courtade-Kumar Most Informative Boolean Function Conjecture and a Symmetrized Li-Médard Conjecture are Equivalent
Leighton Pate Barnes, Ayfer Özgür · 2020
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 Lαnorm of Tpf for 1 ≤ α ≤ 2 where Tpis 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 Lα-norm related quantity can cross zero as a function of α, and show that these two conjectures are essentially equivalent.