To the families of geometric measures of convex bodies (the area measures of Aleksandrov-Fenchel-Jessen, the curvature measures of Federer, and the recently discovered dual curvature measures) a new family is added. The new family of geometric measures, called chord measures, arises from the study of integral geometric invariants of convex bodies. The Minkowski problems for the new measures and their logarithmic variants are proposed and attacked. When the given ‘data’ is sufficiently regular, these problems are a new type of fully nonlinear partial differential equations involving dual quermassintegrals of functions. Major cases of these Minkowski problems are solved without regularity assumptions.

中文翻译：

---

积分几何中的弦测度及其闵可夫斯基问题

凸体几何度量家族（Aleksandrov-Fenchel-Jessen 的面积度量、费德勒的曲率度量以及最近发现的双曲率度量）添加了一个新家族。新的几何测度家族，称为弦测度，源于对凸体的积分几何不变量的研究。提出并攻击了新测度及其对数变体的闵可夫斯基问题。当给定的“数据”足够规则时，这些问题是涉及函数的对偶质量积分的新型完全非线性偏微分方程。这些闵可夫斯基问题的主要情况是在没有规律性假设的情况下解决的。