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Author: Bobkov, S. G. × Author: Houdré, Christian × End date: 1999 × Start date: 1995 × search.filters.applied.f.resourcesubtype: Pre-print ×

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Sharp Constants in Some Multiplicative Sobolev Inequalities

1995-09-16 , Bobkov, S. G. , Houdré, Christian

The optimal constants in the multiplicative Sobolev inequalities where the gradient is estimated in the L_1-norm and the function in two different Lebesgue norms are found. With the optimal constants, these inequalities turn out to still be equivalent to the isoperimetric property of the balls in the Euclidean space. In the course of the proof, relations between Lorentz and Lebesgue spaces are studied (and also applied to some different measures, e.g., Riesz potentials).

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Dimension Free Weak Concentration of Measure Phenomenon

1995-07-24 , Bobkov, S. G. , Houdré, Christian

For product probability measures \mu^n, we obtain necessary and sufficient conditions (in terms of \mu) for dimension free isoperimetric inequalities of the form \mu^n (A + h[-1,1]^n)\ge R_h(\mu^n(A)) to hold; for a function R such that R(p) > p, for all (some) p \in (0,1), and for h > 0 large enough. Some questions related to the concentration of measure phenomenon are also discussed.

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A Characterization of Gaussian Measures via the Isoperimetric Property of Half-Spaces

1995-07-25 , Bobkov, S. G. , Houdré, Christian

If the half-spaces of the form {x\in R^n: x_1 \le c} are extremal in the isoperimetric problem for the product measure \mu^n, n\ge 2, then \mu is Gaussian.

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Variance of Lipschitz Functions and an Isoperimetric Problem for a Class of Product Measures

1995-07-10 , Bobkov, S. G. , Houdré, Christian

The maximal variance of Lipschitz functions (with respect to the \ell_1-distance) of independent random vectors is found. This is then used to solve the isoperimetric problem, uniformly in the class of product probability measures with given variance.

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Isoperimetric Constants for Product Probability Measures

1995-07-24 , Bobkov, S. G. , Houdré, Christian

A dimension free lower bound is found for isoperimetric constants of product probability measures. From this, some analytic inequalities are derived.

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