THE ORLICZ-BRUNN-MINKOWSKI THEORY: A GENERAL FRAMEWORK, ADDITIONS, AND INEQUALITIES
THE ORLICZ-BRUNN-MINKOWSKI THEORY: A GENERAL FRAMEWORK, ADDITIONS, AND INEQUALITIES
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DOI：
10.4310/jdg/1406033976
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发表时间：
2013-01
影响因子：
2.5
通讯作者：
R. Gardner;D. Hug;W. Weil
中科院分区：
文献类型：
--
作者：
R. Gardner;D. Hug;W. Weil
由Lutwak、Yang和Zhang提出的Orlicz-Brunn-Minkowski理论是经典Brunn-Minkowski理论的一个新的推广。它代表了Lp-Brunn-Minkowski理论的推广，类似于Orlicz空间推广Lp空间的方式。对于适当的凸函数'：（0;1）m！（0;1），给出了Rn中任意集合组合的一种新方法.这种运算称为Orlicz加法，用+'表示，它有几个理想的性质，但不是结合的，除非它简化为Lp加法。一个一般的框架介绍了Orlicz-Brunn-Minkowski理论，其中包括新增加的和以前介绍的概念，并首次明确了与Orlicz空间和规范的关系。它还表明，Orlicz加法是密切相关的一个自然的和基本的推广Minkowski加法称为M-加法。得到的结果表明，粗略地说，Orlicz-Brunn-Minkowski理论是最一般的可能的基础上，除了保留所有的基本几何性质享有的Lp-Brunn-Minkowski理论。得到了M-加法和Orlicz加法的Brunn-Minkowski型不等式。新的Orlicz-Brunn-Minkowski不等式蕴涵了Lp-Brunn-Minkowski不等式。得到了新的Orlicz-Minkowski不等式，推广了Lp-Minkowski不等式.其中之一与Lutwak，Yang和Zhang的已证明的log-Brunn-Minkowski不等式有关，实际上这两个不等式一起被证明分裂了经典的Brunn-Minkowski不等式。
The Orlicz-Brunn-Minkowski theory, introduced by Lutwak, Yang, and Zhang, is a new extension of the classical Brunn-Minkowski theory. It represents a generalization of the Lp-Brunn-Minkowski theory, analogous to the way that Orlicz spaces generalize Lp spaces. For appropriate convex functions ' : (0;1) m ! (0;1), a new way of combining arbitrary sets in R n is introduced. This operation, called Orlicz addition and denoted by +', has several desirable properties, but is not associative unless it reduces to Lp addition. A general framework is introduced for the Orlicz-Brunn-Minkowski theory that includes both the new addition and previously introduced concepts, and makes clear for the rst time the relation to Orlicz spaces and norms. It is also shown that Orlicz addition is intimately related to a natural and fundamental generalization of Minkowski addition called M-addition. The results obtained show, roughly speaking, that the Orlicz-Brunn-Minkowski theory is the most general possible based on an addition that retains all the basic geometrical properties enjoyed by the Lp-Brunn-Minkowski theory. Inequalities of the Brunn-Minkowski type are obtained, both for M-addition and Orlicz addition. The new Orlicz-Brunn-Minkowski inequality implies the Lp-Brunn-Minkowski in- equality. New Orlicz-Minkowski inequalities are obtained that generalize the Lp-Minkowski inequality. One of these has connections with the conjectured log-Brunn-Minkowski inequal- ity of Lutwak, Yang, and Zhang, and in fact these two inequalities together are shown to split the classical Brunn-Minkowski inequality.