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

TL;DR

This paper explores the deep connections between social choice theory, computational complexity, Gaussian geometry, and Boolean functions, emphasizing reduction techniques and recent breakthroughs like the Majority Is Stablest Theorem.

Contribution

It synthesizes various mathematical and computational concepts, highlighting the technique of reducing Gaussian inequalities to Boolean functions and applying induction for proofs.

Findings

Unified framework linking social choice and Gaussian geometry

Reduction technique from Gaussian to Boolean functions

Recent proof of the Majority Is Stablest Theorem

Abstract

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.