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 ![m=E[f(X)]](/images/equations/HellingerConjecture/Inline5.svg)
,
and let 
.
With 
denoting the Boolean noise operator, the
conjectured inequality is
![sqrt(1-m^2)-E[sqrt(1-(T_rhof(X))^2)]<=1-sqrt(1-rho^2).](/images/equations/HellingerConjecture/NumberedEquation1.svg) |
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, Proposition 1).
