• Bernoulli
  • Volume 23, Number 2 (2017), 1056-1081.

Perimeters, uniform enlargement and high dimensions

Franck Barthe and Benoît Huou

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We study the isoperimetric problem in product spaces equipped with the uniform distance. Our main result is a characterization of isoperimetric inequalities which, when satisfied on a space, are still valid for the product spaces, up a to a constant which does not depend on the number of factors. Such dimension free bounds have applications to the study of influences of variables.

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Bernoulli, Volume 23, Number 2 (2017), 1056-1081.

Received: December 2014
Revised: September 2015
First available in Project Euclid: 4 February 2017

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influences isoperimetry


Barthe, Franck; Huou, Benoît. Perimeters, uniform enlargement and high dimensions. Bernoulli 23 (2017), no. 2, 1056--1081. doi:10.3150/15-BEJ769.

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  • [1] Ambrosio, L., Fusco, N. and Pallara, D. (2000). Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. New York: The Clarendon Press.
  • [2] Barthe, F. (2002). Log-concave and spherical models in isoperimetry. Geom. Funct. Anal. 12 32–55.
  • [3] Barthe, F. (2004). Infinite dimensional isoperimetric inequalities in product spaces with the supremum distance. J. Theoret. Probab. 17 293–308.
  • [4] Barthe, F., Cattiaux, P. and Roberto, C. (2006). Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoam. 22 993–1067.
  • [5] Bobkov, S.G. (1997). Isoperimetric problem for uniform enlargement. Studia Math. 123 81–95.
  • [6] Bobkov, S.G. and Houdré, C. (1997). Some connections between isoperimetric and Sobolev-type inequalities. Mem. Amer. Math. Soc. 129 viii+111.
  • [7] Bobkov, S.G. and Houdré, C. (2000). Weak dimension-free concentration of measure. Bernoulli 6 621–632.
  • [8] Bollobás, B. and Leader, I. (1991). Edge-isoperimetric inequalities in the grid. Combinatorica 11 299–314.
  • [9] Kahn, J., Kalai, G. and Linial, N. (1988). The influence of variables on Boolean functions. In Proceedings of 29th IEEE Symposium on Foundations of Computer Sciences 68–80.
  • [10] Kalai, G. and Safra, S. (2006). Threshold phenomena and influence: Perspectives from mathematics, computer science, and economics. In Computational Complexity and Statistical Physics. St. Fe Inst. Stud. Sci. Complex. 25–60. New York: Oxford Univ. Press.
  • [11] Keller, N. (2011). On the influences of variables on Boolean functions in product spaces. Combin. Probab. Comput. 20 83–102.
  • [12] Keller, N., Mossel, E. and Sen, A. (2012). Geometric influences. Ann. Probab. 40 1135–1166.
  • [13] Kuczma, M. (2009). An Introduction to the Theory of Functional Equations and Inequalities, 2nd ed. Basel: Birkhäuser. Cauchy’s equation and Jensen’s inequality, Edited and with a preface by Attila Gilányi.
  • [14] Latała, R. and Oleszkiewicz, K. (2000). Between Sobolev and Poincaré. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1745 147–168. Berlin: Springer.
  • [15] Milman, E. (2009). On the role of convexity in functional and isoperimetric inequalities. Proc. Lond. Math. Soc. (3) 99 32–66.
  • [16] Morgan, F. (2006). Isoperimetric estimates in products. Ann. Global Anal. Geom. 30 73–79.
  • [17] Peetre, J. (1968). On interpolation functions. II. Acta Sci. Math. (Szeged) 29 91–92.
  • [18] Ros, A. (2005). The isoperimetric problem. In Global Theory of Minimal Surfaces. Clay Math. Proc. 2 175–209. Providence, RI: Amer. Math. Soc.
  • [19] Talagrand, M. (1991). A new isoperimetric inequality and the concentration of measure phenomenon. In Geometric Aspects of Functional Analysis (198990). Lecture Notes in Math. 1469 94–124. Berlin: Springer.