Social choice, computational complexity, Gaussian geometry, and Boolean functions

Ryan O'Donnell

Published 2014-07-29, updated 2014-08-04Version 2

We describe a web of connections between the following topics: the mathematical theory of voting and social choice; the computational complexity of the Maximum Cut problem; the Gaussian Isoperimetric Inequality and Borell's generalization thereof; the Hypercontractive Inequality of Bonami; and, the analysis of Boolean functions. A major theme is the technique of reducing inequalities about Gaussian functions to inequalities about Boolean functions f : {-1,1}^n -> {-1,1}, and then using induction on n to further reduce to inequalities about functions f : {-1,1} -> {-1,1}. We especially highlight De, Mossel, and Neeman's recent use of this technique to prove the Majority Is Stablest Theorem and Borell's Isoperimetric Inequality simultaneously.