The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem
The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem
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DOI：
10.4310/jdg/1214454097
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发表时间：
1993
影响因子：
2.5
通讯作者：
E. Lutwak
中科院分区：
文献类型：
--
作者：
E. Lutwak
Brunn-Minkowski理论是数量凸性的核心。它的起源在闵可夫斯基的加入他的概念混合卷与布伦-闵可夫斯基不等式。闵可夫斯基的主要贡献之一的理论是，以显示如何这一理论可以发展的几个基本概念：支持功能，闵可夫斯基组合，和混合卷。30年前，Firey [8]（参见Burago和Zalgaller [4，§24.6]）扩展了Minkowski组合的概念，并对每个真实的p > 1引入了他所谓的p-和。这一系列文章的目的是表明，这些火的组合导致一个布伦-闵可夫斯基理论的每一个p > 1。设Jf表示欧氏λ z-空间R中的凸体（内部非空的紧凸子集）集。设3E”表示在其内部包含原点的凸体的集合。为了K e 3？设hκ = h（K，·）：S ~ -> R表示K |的支撑函数，即，对u ∈ S~，hκ{u）= h（K，k）= max{wx：xeK}，其中u-x表示R中的标准内积.集合Jf将被视为配备有通常的豪斯多夫度量d，由d（K，L）= \hκ AJ^定义，其中|1^是单位球面C（S~）上连续函数空间的sup（或max）范数。对于K，L e J e，α，β > 0（不都为零），Minkowski线性组合aK + βL e J e？限定
The Brunn-Minkowski theory is the heart of quantitative convexity. It had its origins in Minkowski's joining his notion of mixed volumes with the Brunn-Minkowski inequality. One of Minkowski's major contributions to the theory was to show how this theory could be developed from a few basic concepts: support functions, Minkowski combinations, and mixed volumes. Thirty years ago, Firey [8] (see Burago and Zalgaller [4, §24.6]) extended the notion of a Minkowski combination, and introduced, for each real p > 1, what he called p-sums. It is the aim of this series of articles to show that these Firey combinations lead to a Brunn-Minkowski theory for each p > 1. Let Jf denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean λz-space, R . Let 3£" denote the set of convex bodies containing the origin in their interiors. For K e 3? , let hκ = h(K, •): S ~ -> R denote the support function of K\ i.e., for u e S~, hκ{u) = h(K, ύ) = max{w x : x e K} , where u-x denotes the standard inner product in R . The set Jf will be viewed as equipped with the usual Hausdorff metric, d, defined by d(K, L) = \hκ AJ^ , where | 1^ is the sup (or max) norm on the space of continuous functions on the unit sphere, C(S~). For K, L e J£ , and α, β > 0 (not both zero), the Minkowski linear combination aK + βL e J? is defined by