论文标题
亚历山德罗夫 - 芬渠道不等式的凸多属性的极端
The Extremals of the Alexandrov-Fenchel Inequality for Convex Polytopes
论文作者
论文摘要
Alexandrov-fenchel不平等，是对任意混合体积的经典等级不等式的深远概括，位于凸几何的核心。其极端身体的特征是一个长期的开放问题，可以追溯到亚历山德罗夫（Alexandrov）的1937年原始纸。已知的极端人已经构成了一个非常富裕的家庭，甚至由于施耐德（Schneider）而对其一般结构的基本猜想都不完整。在本文中，我们完全解决了亚历山德罗夫 - 芬德漏斗不等式的极端凸多属性。特别是，我们表明极端是源于三种不同机制的组合：翻译，支持和维度。这些机制的表征需要开发多种技术，从而为非平滑凸体的混合体积的几何形状提供了新的启示。我们的主要结果以多种方式延伸了多面体，包括设置任意凸体的Quermassintegrals。作为我们主要结果的应用，我们解决了斯坦利的问题，介绍了在部分有序集的组合中出现的某些对数凸线序列的极端行为。
The Alexandrov-Fenchel inequality, a far-reaching generalization of the classical isoperimetric inequality to arbitrary mixed volumes, lies at the heart of convex geometry. The characterization of its extremal bodies is a long-standing open problem that dates back to Alexandrov's original 1937 paper. The known extremals already form a very rich family, and even the fundamental conjectures on their general structure, due to Schneider, are incomplete. In this paper, we completely settle the extremals of the Alexandrov-Fenchel inequality for convex polytopes. In particular, we show that the extremals arise from the combination of three distinct mechanisms: translation, support, and dimensionality. The characterization of these mechanisms requires the development of a diverse range of techniques that shed new light on the geometry of mixed volumes of nonsmooth convex bodies. Our main result extends further beyond polytopes in a number of ways, including to the setting of quermassintegrals of arbitrary convex bodies. As an application of our main result, we settle a question of Stanley on the extremal behavior of certain log-concave sequences that arise in the combinatorics of partially ordered sets.
