论文标题
凸几何形状中的集合估计值
Sumset estimates in convex geometry
论文作者
论文摘要
集合估计值在组中有限集的总和的基数构成了添加剂组合学工具包的重要组成部分。近年来，引入了许多此类不平等的概率或熵类似物。我们在$ {\ mathbb r}^n $上研究了这些集合估计值的类似物和lebesgue测量。首先，我们观察到，关于Minkowski求和，体积是凸体空间上任意顺序的超模型。其次，我们探讨了plünnecke-ruzsa不平等变体的凸几何类似物中的尖锐常数。在本文的最后一部分中，我们提供了这些不平等现象与经典的罗杰斯·夏德（Rogers-Shephard）不平等现象的联系。
Sumset estimates, which provide bounds on the cardinality of sumsets of finite sets in a group, form an essential part of the toolkit of additive combinatorics. In recent years, probabilistic or entropic analogs of many of these inequalities were introduced. We study analogues of these sumset estimates in the context of convex geometry and Lebesgue measure on ${\mathbb R}^n$. First, we observe that, with respect to Minkowski summation, volume is supermodular to arbitrary order on the space of convex bodies. Second, we explore sharp constants in the convex geometry analogues of variants of the Plünnecke-Ruzsa inequalities. In the last section of the paper, we provide connections of these inequalities to the classical Rogers-Shephard inequality.
