论文标题
在与Prékopa-Leindler不等式有关的功能的几何组合中
On a geometric combination of functions related to Prékopa-Leindler inequality
论文作者
论文摘要
我们在$ \ mathbb {r} ^n $上引入了一个新的操作，我们称为几何组合；它是通过大规模运输方法获得的，可以使用反向分布功能。该操作的主要特征是几何组合的Lebesgue积分等于两个单独的积分的几何平均值。自然的结果，我们得出了Prékopa-Leindler类型的新功能不平等。当应用于两个可测量集的特征函数时，它们的几何组合提供了一个集合，其体积等于两个单独卷的几何平均值。在凸面的框架内，通过将几何组合与$ 0 $ -SUM进行比较，我们获得了log-brunn-minkowski不平等的替代证明，用于无条件的凸形主体和具有$ n $对称性的凸形物体。
We introduce a new operation between nonnegative integrable functions on $\mathbb{R} ^n$, that we call geometric combination; it is obtained via a mass transportation approach, playing with inverse distribution functions. The main feature of this operation is that the Lebesgue integral of the geometric combination equals the geometric mean of the two separate integrals; as a natural consequence, we derive a new functional inequality of Prékopa-Leindler type. When applied to the characteristic functions of two measurable sets, their geometric combination provides a set whose volume equals the geometric mean of the two separate volumes. In the framework of convex bodies, by comparing the geometric combination with the $0$-sum, we get an alternative proof of the log-Brunn-Minkowski inequality for unconditional convex bodies and for convex bodies with $n$ symmetries.
