The Annals of Probability

Geometric influences

Nathan Keller, Elchanan Mossel, and Arnab Sen

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We present a new definition of influences in product spaces of continuous distributions. Our definition is geometric, and for monotone sets it is identical with the measure of the boundary with respect to uniform enlargement. We prove analogs of the Kahn–Kalai–Linial (KKL) and Talagrand’s influence sum bounds for the new definition. We further prove an analog of a result of Friedgut showing that sets with small “influence sum” are essentially determined by a small number of coordinates. In particular, we establish the following tight analog of the KKL bound: for any set in ℝn of Gaussian measure t, there exists a coordinate i such that the ith geometric influence of the set is at least $ct(1-t)\sqrt{\log n}/n$, where c is a universal constant. This result is then used to obtain an isoperimetric inequality for the Gaussian measure on ℝn and the class of sets invariant under transitive permutation group of the coordinates.

Article information

Ann. Probab., Volume 40, Number 3 (2012), 1135-1166.

First available in Project Euclid: 4 May 2012

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 05D40: Probabilistic methods

Influences product space Kahn–Kalai–Linial influence bound Gaussian measure isoperimetric inequality


Keller, Nathan; Mossel, Elchanan; Sen, Arnab. Geometric influences. Ann. Probab. 40 (2012), no. 3, 1135--1166. doi:10.1214/11-AOP643.

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  • [1] Bakry, D. and Ledoux, M. (1996). Lévy–Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator. Invent. Math. 123 259–281.
  • [2] Barthe, F. (2004). Infinite dimensional isoperimetric inequalities in product spaces with the supremum distance. J. Theoret. Probab. 17 293–308.
  • [3] Bobkov, S. (1996). Extremal properties of half-spaces for log-concave distributions. Ann. Probab. 24 35–48.
  • [4] Bobkov, S. G. (1997). Isoperimetric problem for uniform enlargement. Studia Math. 123 81–95.
  • [5] Borell, C. (1975). The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30 207–216.
  • [6] Bourgain, J., Kahn, J., Kalai, G., Katznelson, Y. and Linial, N. (1992). The influence of variables in product spaces. Israel J. Math. 77 55–64.
  • [7] Ehrhard, A. (1983). Symétrisation dans l’espace de Gauss. Math. Scand. 53 281–301.
  • [8] Friedgut, E. (1998). Boolean functions with low average sensitivity depend on few coordinates. Combinatorica 18 27–35.
  • [9] Friedgut, E. and Kalai, G. (1996). Every monotone graph property has a sharp threshold. Proc. Amer. Math. Soc. 124 2993–3002.
  • [10] Graham, B. T. and Grimmett, G. R. (2006). Influence and sharp-threshold theorems for monotonic measures. Ann. Probab. 34 1726–1745.
  • [11] Hatami, H. (2009). Decision trees and influences of variables over product probability spaces. Combin. Probab. Comput. 18 357–369.
  • [12] Kahn, J., Kalai, G. and Linial, N. The influence of variables on Boolean functions. In Proceedings of 29th IEEE Symp. Foundations of Computer Science (FOCS, 1988).
  • [13] Kalai, G. and Safra, S. (2006). Threshold phenomena and influence: Perspectives from mathematics, computer science, and economics. In Computational Complexity and Statistical Physics 25–60. Oxford Univ. Press, New York.
  • [14] Keller, N. (2011). On the influences of variables on Boolean functions in product spaces. Combin. Probab. Comput. 20 83–102.
  • [15] Margulis, G. A. (1974). Probabilistic characteristics of graphs with large connectivity. Problemy Peredachi Informatsii 10 101–108.
  • [16] Mossel, E., O’Donnell, R. and Oleszkiewicz, K. (2010). Noise stability of functions with low influences: Invariance and optimality. Ann. of Math. (2) 171 295–341.
  • [17] Neyman, J. and Pearson, E. S. (1933). On the problem of the most efficient tests of statistical hypotheses. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 231 289–337.
  • [18] Russo, L. (1982). An approximate zero–one law. Z. Wahrsch. Verw. Gebiete 61 129–139.
  • [19] Steele, J. M. (1986). An Efron–Stein inequality for nonsymmetric statistics. Ann. Statist. 14 753–758.
  • [20] Sudakov, V. N. and Tsirelson, B. (1974). Extremal properties of half-spaces for spherically invariant measures. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 14–24, 165.
  • [21] Talagrand, M. (1994). On Russo’s approximate zero–one law. Ann. Probab. 22 1576–1587.
  • [22] Zajíček, L. (1987/88). Porosity and σ-porosity. Real Anal. Exchange 13 314–350.