Convex Influences
Abstract
We introduce a new notion of influence for symmetric convex sets over Gaussian space, which we term "convex influence". We show that this new notion of influence shares many of the familiar properties of influences of variables for monotone Boolean functions $f: \{\pm1\}^n \to \{\pm1\}.$ Our main results for convex influences give Gaussian space analogues of many important results on influences for monotone Boolean functions. These include (robust) characterizations of extremal functions, the Poincaré inequality, the KahnKalaiLinial theorem, a sharp threshold theorem of Kalai, a stability version of the KruskalKatona theorem due to O'Donnell and Wimmer, and some partial results towards a Gaussian space analogue of Friedgut's junta theorem. The proofs of our results for convex influences use very different techniques than the analogous proofs for Boolean influences over $\{\pm1\}^n$. Taken as a whole, our results extend the emerging analogy between symmetric convex sets in Gaussian space and monotone Boolean functions from $\{\pm1\}^n$ to $\{\pm1\}$
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.03107
 Bibcode:
 2021arXiv210903107D
 Keywords:

 Computer Science  Computational Complexity;
 Mathematics  Combinatorics;
 Mathematics  Probability
 EPrint:
 32 pages