论文标题
加权Brunn-Minkowski理论I I：在加权表面积测量中
Weighted Brunn-Minkowski Theory I: On Weighted Surface Area Measures
论文作者
论文摘要
凸数学中的Brunn-Minkowski理论涉及凸体的体积，混合体积和表面积测量。我们研究了这些概念对$ \ mathbb {r}^n $中密度的概括 - 尤其是混合体积的加权版本（所谓的混合措施），当时多达三个不同的凸面。然后，我们制定和分析经典表面积测量的加权版本，并获得三个物体混合度量的新积分公式。作为一种应用，我们证明了旋转不变的对数孔尺寸的贝佐特型不平等，这是Artstein-Avidan，Florentin和Ostrover的概括。即使对于标准高斯措施的特殊情况，结果也是新的和有趣的。
The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed volumes, and surface area measures of convex bodies. We study generalizations of these concepts to Borel measures with density in $\mathbb{R}^n$-- in particular, the weighted versions of mixed volumes (the so-called mixed measures) when dealing with up to three distinct convex bodies. We then formulate and analyze weighted versions of classical surface area measures, and obtain a new integral formula for the mixed measure of three bodies. As an application, we prove a Bézout-type inequality for rotational invariant log-concave measures, generalizing a result by Artstein-Avidan, Florentin and Ostrover. The results are new and interesting even for the special case of the standard Gaussian measure.
