论文标题
关于翻译不变估值和Aleksandrov-Fenchel不平等的混合杂货关系
On mixed Hodge-Riemann relations for translation-invariant valuations and Aleksandrov-Fenchel inequalities
论文作者
论文摘要
最近，在几个特殊情况下，最初命名的作者在几个特殊情况下猜想并证明了霍迪·里曼（Hodge-Riemann）关系的一种版本。 Lefschetz操作员认为，在那里出现了几个欧几里得球的混合体积的产品或卷积。在这里，我们证明，在（共同）一度，如果球被平滑边界和正式曲率的几个不同（中央对称的）凸体代替，则霍奇 - 里曼的关系持续存在。尽管这些混合的Hodge-Riemann关系的卷积关系直接暗示了Aleksandrov-Fenchel的不平等，但它们屈服于产品的双重操作是一种新的不平等。这种新的不平等增强了较低维凸的Aleksandrov-Fenchel不平等的经典后果，并概括了S. Alesker最近发现的一些几何不平等现象
A version of the Hodge-Riemann relations for valuations was recently conjectured and proved in several special cases by the first-named author. The Lefschetz operator considered there arises as either the product or the convolution with the mixed volume of several Euclidean balls. Here we prove that in (co-)degree one the Hodge-Riemann relations persist if the balls are replaced by several different (centrally symmetric) convex bodies with smooth boundary with positive Gauss curvature. While these mixed Hodge-Riemann relations for the convolution directly imply the Aleksandrov-Fenchel inequality, they yield for the dual operation of the product a new inequality. This new inequality strengthens classical consequences of the Aleksandrov-Fenchel inequality for lower dimensional convex bodies and generalizes some of the geometric inequalities recently discovered by S. Alesker
