Hellinger Conjecture

The Hellinger conjecture is the assertion that a single coordinate maximizes a particular loss of Hellinger affinity for a Boolean function under a Boolean noise operator (Anantharam et al. 2017). For and , let have the uniform distribution on , put , and let . With denoting the Boolean noise operator, the conjectured inequality is

It allows arbitrary output bias , and equality holds for .

For a nonconstant , let be the output of the binary symmetric channel with crossover probability , and let and be its conditional distributions given and . The left side is , where is Hellinger distance. The inequality implies the Courtade-Kumar conjecture (Anantharam et al. 2017, p. 2).