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Convex Influences

Authors Anindya De, Shivam Nadimpalli, Rocco A. Servedio

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Author Details

Anindya De
  • University of Pennsylvania, Philadelphia, PA, USA
Shivam Nadimpalli
  • Columbia University, New York, NY, USA
Rocco A. Servedio
  • Columbia University, New York, NY, USA


This work was done while A. D. was participating in the "Probability, Geometry, and Computation in High Dimensions" program at the Simons Institute for the Theory of Computing. A. D. wishes to thank the Simons Institute for their support.

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Anindya De, Shivam Nadimpalli, and Rocco A. Servedio. Convex Influences. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 53:1-53:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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: {±1}ⁿ → {±1}. 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 Kahn-Kalai-Linial theorem [J. Kahn et al., 1988], a sharp threshold theorem of Kalai [G. Kalai, 2004], a stability version of the Kruskal-Katona theorem due to O'Donnell and Wimmer [R. O'Donnell and K. Wimmer, 2009], and some partial results towards a Gaussian space analogue of Friedgut’s junta theorem [E. Friedgut, 1998]. The proofs of our results for convex influences use very different techniques than the analogous proofs for Boolean influences over {±1}ⁿ. Taken as a whole, our results extend the emerging analogy between symmetric convex sets in Gaussian space and monotone Boolean functions from {±1}ⁿ to {±1}.

Subject Classification

ACM Subject Classification
  • Mathematics of computing
  • Mathematics of computing → Probability and statistics
  • Fourier analysis of Boolean functions
  • convex geometry
  • influences
  • threshold phenomena


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