Dimension Free Weak Concentration of Measure Phenomenon

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.ISF NXZ000, NXZ300 and Grant No. 94-1.4-57 from the Grant Center of Fundamental Sciences in Sankt-Petersburg Universit