论文标题
表面积测量对数凸线功能的测量
Surface area measures of log-concave functions
论文作者
论文摘要
本文的起源在两篇论文中：一篇是Colesanti和Fragalà研究对数凸函数的表面积测量，另一篇是Cordero-erausquin和Klartag关于凸功能的矩度的。这些概念是相同的，在本文中，我们继续研究相同的结构及其概括。 在上半年，我们证明了在最小和最佳条件下对数符号函数积分的第一个变化公式。我们还解释了为什么这种结果是从上述论文中对两个已知定理的普遍概括。 在下半年，我们将功能表面积测量的定义扩展到了l^p-设置，从而推广了Lutwak的经典定义。在这种广义的环境中，我们证明了偶数措施的功能性Minkowski存在定理。这是Cordero-erausquin和Klartag定理的部分扩展，该定理处理了Case P = 1的情况，甚至不一定是措施。
This paper's origins are in two papers: One by Colesanti and Fragalà studying the surface area measure of a log-concave function, and one by Cordero-Erausquin and Klartag regarding the moment measure of a convex function. These notions are the same, and in this paper we continue studying the same construction as well as its generalization. In the first half the paper we prove a first variation formula for the integral of log-concave functions under minimal and optimal conditions. We also explain why this result is a common generalization of two known theorems from the above papers. In the second half we extend the definition of the functional surface area measure to the L^p-setting, generalizing a classic definition of Lutwak. In this generalized setting we prove a functional Minkowski existence theorem for even measures. This is a partial extension of a theorem of Cordero-Erausquin and Klartag that handled the case p=1 for not necessarily even measures.
