论文标题
关于翻译不变估值的Hodge-Riemann关系
On Hodge-Riemann relations for translation-invariant valuations
论文作者
论文摘要
Alesker产品将凸体的平滑翻译不变估值的空间转化为可交换的关联Unital代数，使Poincaré二元性和硬Lefschetz定理感到满意。在本文中，猜想了Alesker代数的Hodge-Riemann关系版本，并在两种特殊情况下证明了猜想：甚至是估值，以及1个同质的估值。然后将后一个结果推断出Aleksandrov-Fenchel不平等的特殊情况。最后，猜想了硬Lefschetz定理的混合版本和Hodge-Riemann关系，并且表明Aleksandrov-fenchel不平等的不平等是后者的全部一般性。
The Alesker product turns the space of smooth translation-invariant valuations on convex bodies into a commutative associative unital algebra, satisfying Poincaré duality and the hard Lefschetz theorem. In this article, a version of the Hodge-Riemann relations for the Alesker algebra is conjectured, and the conjecture is proved in two particular situations: for even valuations, and for 1-homogeneous valuations. The latter result is then used to deduce a special case of the Aleksandrov-Fenchel inequality. Finally, mixed versions of the hard Lefschetz theorem and of the Hodge-Riemann relations are conjectured, and it is shown that the Aleksandrov-Fenchel inequality follows from the latter in its full generality.
