On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation
On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation
复制标题
DOI：
10.1007/978-3-642-55925-9_36
复制
发表时间：
1976-08
影响因子：
1.7
通讯作者：
H. J. Brascamp;E. Lieb
中科院分区：
文献类型：
--
作者：
H. J. Brascamp;E. Lieb
我们将 Prékopa-Leindler 定理扩展到其他类型的两个正函数的凸组合，并通过引入本质加法的概念来加强 Prékopa-Leindler 和 Brunn-Minkowski 定理。我们对 Prékopa-Leindler 定理的证明比原来的要简单。我们锐化了对数凹函数的边际为对数凹的不等式，并证明了此类函数的各种矩不等式。最后，我们使用这些结果推导具有凸势的扩散方程的基本解的不等式。
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.