From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. (English) Zbl 0969.26019

This is an interesting paper with several applications. To summarize the results tersely the best way is recalling the abstract of the paper:
“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”.


26D15 Inequalities for sums, series and integrals
52A40 Inequalities and extremum problems involving convexity in convex geometry
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