A Concentration of Measure Phenomenon on Uniformly Convex Bodies
In his paper [M] B. Maurey gave a very simple proof of the gaussian deviation inequality, based on an inequality of A. Prekopa [Pr] and L. Leindler [L]. Motivated by Talagrand’s isoperimetric inequality for the cube (cf. [T.3]) Maurey defined a property which he called property (τ). He proved that in the gaussian case this property is a consequence of the Prekopa-Leindler inequality. This method can be easily extended to give a concentration of measure phenomenon for a particular measure associated with uniformly convex spaces whose modulus of convexity satisfies a certain condition. In [Pi] G. Pisier proved that this condition can be guaranteed after renorming. Prom this we deduce the concentration of measure result of M. Gromov and V.D. Milman (cf. [G.M.2]) for uniformly convex bodies, whose modulus of convexity satisfies the condition mentioned above.
KeywordsUnit Ball Convex Body Lipschitz Function Convex Space Measurable Subset
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