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On a reverse form of the Brascamp-Lieb inequality. (English) Zbl 0901.26010

We derive a reverse form of the Brascamp-Lieb inequalities. The proof uses convex solutions of the Monge-Ampère equation and provides the Brascamp-Lieb inequalities and the converse inequalities altogether. We study equality cases for functions of one real variable. Next, we give applications to convex geometry: volume estimates, Brunn-Minkowski type inequalities.

MSC:

26D15 Inequalities for sums, series and integrals
52A40 Inequalities and extremum problems involving convexity in convex geometry
52A38 Length, area, volume and convex sets (aspects of convex geometry)
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