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Inequalities pp 441-464 | Cite as

On Extensions of the Brunn-Minkowski and Prékopa-Leindler Theorems, Including Inequalities for Log Concave Functions, and with an Application to the Diffusion Equation

  • Herm Jan Brascamp
  • Elliott H. Lieb

Abstract

We extend the Prékopa-Leindler theorem to other types of convex combinations of two positive functions and we strengthen the Prékopa—Leindler and Brunn-Minkowski theorems by introducing the notion of essential addition. Our proof of the Prékopa—Leindler theorem is simpler than the original one. We sharpen the inequality that the marginal of a log concave function is log concave, and we prove various moment inequalities for such functions. Finally, we use these results to derive inequalities for the fundamental solution of the diffusion equation with a convex potential.

Keywords

Diffusion Equation Fundamental Solution Moment Inequality Nonnegative Measurable Function Convex Potential 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Herm Jan Brascamp
    • 1
  • Elliott H. Lieb
    • 2
  1. 1.Department of PhysicsPrinceton UniversityPrinceton
  2. 2.Department of Mathematics and PhysicsPrinceton UniversityPrinceton

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