On Extensions of the Brunn-Minkowski and Prékopa-Leindler Theorems, Including Inequalities for Log Concave Functions, and with an Application to the Diffusion Equation
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.
KeywordsDiffusion Equation Fundamental Solution Moment Inequality Nonnegative Measurable Function Convex Potential
Unable to display preview. Download preview PDF.
- 1.L. Lusternik, Die Brunn-Minkowskische Ungleichung für beliebige messbare Mengen, C. R. Dokl. Acad. Sci. URSS No. 3, 8 (1935), 55–58.Google Scholar
- 6.H. J. Brascamp and E. H. Lieb, Some inequalities for Gaussian measures, in “Functional Integral and its Applications” (A. M. Arthurs, Ed.), Clarendon Press, Oxford, 1975.Google Scholar
- 7.H. J. Brascamp and E. H. Lieb, Best constants in Young’s inequality, its converse and its generalization to more than three functions, Advances in Math. 20 (1976).Google Scholar
- 8.Y. Rinott, On convexity of measures, Thesis, Weizmann Institute, Rehovot, Israel, November 1973, to appear.Google Scholar