We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probability measures which satisfy these inequalities 1999 Academic Press

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We develop several applications of the Brunn-Minkowski inequality in the Prékopa-Leindler form. In particular, we show that an argument of B. Maurey may be adapted to deduce from the Prékopa-Leindler theorem the Brascamp-Lieb inequality for strictly convex potentials. We deduce similarly the logarithmic Sobolev inequality for uniformly convex potentials for which we deal more generally with arbitrary norms and obtain some new results in this context. Applications to transportation cost and to concentration on uniformly convex bodies complete the exposition.

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