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

In: Inequalities

Author

Listed:
  • Herm Jan Brascamp

    (Princeton University, Department of Physics)

  • Elliott H. Lieb

    (Princeton University, Department of Mathematics and Physics)

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.

Suggested Citation

  • Herm Jan Brascamp & Elliott H. Lieb, 2002. "On Extensions of the Brunn-Minkowski and Prékopa-Leindler Theorems, Including Inequalities for Log Concave Functions, and with an Application to the Diffusion Equation," Springer Books, in: Michael Loss & Mary Beth Ruskai (ed.), Inequalities, pages 441-464, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-55925-9_36
    DOI: 10.1007/978-3-642-55925-9_36
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