Structure Theory of Infinite Dimensional Banach Spaces and a Gaussian Correlation Problem

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9706828 Schlumprecht The concept of distortion played a crucial role in recent discoveries of infinite dimensional Banach space theory. It tries to specify the following problem: Given the norm topology of a Banach space, how different are the equivalent norms generating this topology? Our main question is the following: Are there any Banach spaces which are distortable but not arbitrarily distortable? Directly connected to the concept of distortion is the concept of stabilization and the spectrum of Lipschitz functions on the sphere of an infinite dimensional Banach space. It was introduced by V. Milman in 1971. The author intends to discuss the question whether, after stabilization, the spectrum of a Lipschitz function must be a connected set. Furthermore it is intended to work on several problems concerning the asymptotic structure of Banach spaces. They could be summed up as follows: How much information about the infinite dimensional properties of a Banach space can be deduced from its asymptotic properties? The second part of this proposal treats the following problem about Gaussian measures: Are two n-dimensional convex symmetric sets positively correlated with respect to the standard Gaussian measure? Banach spaces were introduced to provide a frame in which solutions of Differential Equations exist, i.e. equations which model dynamic systems. The author intends to continue his study about their structure. Roughly speaking, one might summarize the problems as follows: What are the building blocks of Banach spaces, are there "atomic" spaces? The second part of the proposal treats a group of questions concerning multi-dimensional normal distributions. The following question is a generic problem in statistics: One has tables for standard normal distr ibutions, but no tables for multi-dimensional normal distributions (there are simply too many of them). So, if we are dealing with a multi-dimensional problem for which the data can be approximated by a multi-dimensional normal, how can we deal with it? In order to find estimates of confidence regions the author intends to find lower estimates for probability of multi-dimensional events using the one dimensional standard distribution.