The Gauss Image Problem
The Gauss Image Problem
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DOI：
10.1002/cpa.21898
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发表时间：
2020-05-06
影响因子：
3
通讯作者：
Zhao, Yiming
中科院分区：
文献类型：
--
作者：
Boroczky, Karoly J.;Lutwak, Erwin;Zhao, Yiming
布吕恩-闵可夫斯基理论和对偶布吕恩-闵可夫斯基理论是凸几何分析中的两个核心理论，主要研究与凸体相关的全局几何不变量和几何测度。这两种理论在概念上表现出惊人的对偶性，涉及几何和分析中的许多对偶概念，如泛函分析中的对偶空间、凸几何中的极性、几何层析成像中的投影和相交；参见Schneider[49，第507页]的清晰解释。在概念对偶中，欧几里得n空间中凸体K的径向高斯像K（定义如下）扮演了中心角色。径向高斯像是rn在单位球s1上的映射，其值是单位球的子集。众所周知，Aleksandrov在s1上的积分曲率和球面勒贝格测量是通过径向高斯像“联系”起来的，Aleksandrov- fencher - jessen和Federer的经典表面积测量也是如此。n1/th曲率测度（见Schneider[49]定理4.2）。[3]和[27])。径向高斯图像的重要性在最近的工作b[27]中得到了更明显的体现，在b[27]中，人们揭示了长期寻求的对偶曲率测量（费德勒曲率测量的对偶对应）。1996年，通过对径向高斯像的关键利用，在布伦-闵可夫斯基理论和对偶布伦-闵可夫斯基理论之间建立了新的联系。由于这些新的几何度量是通过径向高斯图像来定义的，它变得
The Brunn-Minkowski theory and the dual Brunn-Minkowski theory are two core theories in convex geometric analysis that center on the investigation of global geometric invariants and geometric measures associated with convex bodies. The two theories display an amazing conceptual duality that involves many dual concepts in both geometry and analysis such as dual spaces in functional analysis, polarity in convex geometry, and projection and intersection in geometric tomography; see Schneider [49, p. 507] for a lucid explanation. In the conceptual duality, a central role is assumed by the radial Gauss image K (defined immediately below) of a convex body K in euclidean n-space, R n. The radial Gauss image is a map on the unit sphere, S n1, of R n whose values are subsets of the unit sphere. It is known that Aleksandrov’s integral curvature on S n1 and spherical Lebesgue measure are “linked” via the radial Gauss image, and so are the classical surface area measure of Aleksandrov-Fenchel-Jessen and Federer’s. n1/th curvature measure (see Schneider [49, theorem 4.2. 3] and [27]). The importance of the radial Gauss image was made more evident in the recent work [27], in which the long-sought dual curvature measures (the dual counterparts of Federer’s curvature measures) were unveiled. In [27] new links were established between the Brunn-Minkowski theory and the dual Brunn-Minkowski theory by making critical use of the radial Gauss image. Motivated by the manner in which these new geometric measures are defined via the radial Gauss image, it becomes