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Concentration InequalitiesA Nonasymptotic Theory of Independence$
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Stéphane Boucheron, Gábor Lugosi, and Pascal Massart

Print publication date: 2013

Print ISBN-13: 9780199535255

Published to Oxford Scholarship Online: May 2013

DOI: 10.1093/acprof:oso/9780199535255.001.0001

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Isoperimetry on the Hypercube and Gaussian Spaces

Isoperimetry on the Hypercube and Gaussian Spaces

Chapter:
(p.290) 10 Isoperimetry on the Hypercube and Gaussian Spaces
Source:
Concentration Inequalities
Author(s):

Stéphane Boucheron

Gábor Lugosi

Pascal Massart

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199535255.003.0010

The purpose of this chapter is to explore further the rich connection between concentration and isoperimetry on the n-dimensional binary cube and also on R n , equipped by the canonical Gaussian measure. We introduce an alternative way of measuring the size of the boundary of a subset of the binary hypercube and prove a corresponding isoperimetric inequality. This isoperimetric result is the consequence of Bobkov’s inequality. One of the most important corollaries of Bobkov’s inequality is the Gaussian isoperimetric theorem. We also derive some further results on threshold widths for certain monotone sets.

Keywords:   Gaussian isoperimetric theorem, Bobkov’s inequality, Margulis’ graph connectivity theorem

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