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#### Sharp Constants in Some Multiplicative Sobolev Inequalities

(Georgia Institute of Technology, 1995-09-16)

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 ...

#### Dimension Free Weak Concentration of Measure Phenomenon

(Georgia Institute of Technology, 1995-07-24)

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 ...

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

(Georgia Institute of Technology, 1995-07-25)

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.

#### Isoperimetric Constants for Product Probability Measures

(Georgia Institute of Technology, 1995-07-24)

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

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

(Georgia Institute of Technology, 1995-07-10)

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 ...