优化测度、Bernoulli过程和余型的构造
Constructions of Majorizing Measures, Bernoulli processes and cotype
摘要：
我们提出了三种方法来构建优化措施，在各种设置。这些方法是基于直接建设的增加序列的分区通过一个简单的穷举过程，而不是建设分离的超度量子空间。第一个方案的建设提供了一个简单的统一证明的优测度定理高斯过程和以下事实。如果$A,B$是向量空间中的平衡凸集，并且如果$A$是充分凸的，则对所有$\varepsilon>0$的覆盖数$N(A,\varepsilon B)$的控制意味着在$A$上存在（先验更强的）优化测度，该优化测度提供由$B$引起的距离。这显然是第一次建立了几何和优化措施之间的明确联系，并推广了希尔伯特空间中椭球上优化措施的早期结果，这些结果是通过特定方法获得的。其余大部分的文件是关注的结构有界伯努利（=Radmacher）过程。其结构上的主要猜想是重新制定了几种方式，这是等价的，并等价于存在某些优化措施。针对这一问题，提出了两种优化措施的构造方案。一个允许描述伯努利过程时的指标集，提供了上确界规范，是足够小的。另一个允许证明一个弱形式的主要猜想。
英文摘要：
We present three methods to construct majorizing measures in various settings. These methods are based on direct constructions of increasing sequences of partitions through a simple exhaustion procedure rather than on the construction of well separated ultrametric subspaces. The first scheme of construction provides a simple unified proof of the Majorizing Measure Theorem for Gaussian processes and of the following fact. If $A,B$ are balanced convex sets in a vector space, and if $A$ is sufficiently convex, a control of the covering numbers $N(A,\varepsilon B)$ for all $\varepsilon>0$ implies the (a priori stronger) existence of a majorizing measure on $A$ provided with the distance induced by $B$. This establishes, apparently for the first time, a clear link between geometry and majorizing measures, and generalizes the earlier results on majorizing measures on ellipsoids in Hilbert space, that were obtained by specific methods. Much of the rest of the paper is concerned with the structure of bounded Bernoulli (=Radmacher) processes. The main conjecture on their structure is reformulated in several ways, that are shown to be equivalent, and to be equivalent to the existence of certain majorizing measures. Two schemes of construction of majorizing measures related to this problem are presented. One allows to describe Bernoulli processes when the index set, provided with the supremum norm, is sufficiently small. The other allows to prove a weak form of the main conjecture.
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