On the Optimality of Dictator Functions and Isoperimetric Inequalities on Boolean Hypercubes

In this paper, we present a family of conjectures on the optimality of the dictator function among all Boolean functions for a new family of
$\Phi$
-entropies. Our main theorem shows that there is an ordering of these conjectures, in that if the conjecture is established for a value
$\alpha$
, then it holds for any
$\beta$
:
$\beta\geq \alpha$
• For the parameter range
$\frac{1}{2}\leq\alpha, \beta < 1$
, this family of conjectures is stronger than the mutual information conjecture, originally proposed in a paper by Courtade and Kumar. By considering a limiting value of the noise parameter
$\rho$
, we show how these conjectures relate to isoperimetric inequalities on the Boolean Hypercube of a flavor first considered by Talagrand and later by Kahn and Park. Finally, we obtain bounds to our conjecture using ideas from the proofs of isoperimetric inequalities.
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