Perimeters, uniform enlargement and high dimensions
Franck Barthe, Benoit Huou
TL;DR
This paper investigates the isoperimetric problem in product spaces with uniform distance, providing dimension-free bounds that have implications for variable influence analysis.
Contribution
It characterizes when isoperimetric inequalities in a space extend to product spaces with a uniform distance, independent of the number of factors.
Findings
Dimension-free isoperimetric bounds established
Characterization of inequalities preserved under product formation
Applications to influence of variables in high dimensions
Abstract
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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