Thomas Schlumprecht

An analytic solution to the Busemann-Petty problem on sections of convex bodies

We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in R n with the Fourier transform of powers of the radial function of the body. A parallel section function (or (ni 1)dimensional X-ray) gives the ((ni 1)-dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in R n and leads to a unifled analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies inR n such that the ((ni 1)-dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (n-dimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the Busemann-Petty problem, intersection bodies, and positive deflnite distributions, our formula shows that the answer to the problem depends on the behavior of the (ni 2)-nd derivative of the parallel section functions. The a‐rmative answer to the Busemann-Petty problem for n• 4 and the negative answer for n‚ 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.

An arbitrarily distortable Banach space

In this work we construct a “Tsirelson like Banach space” which is arbitrarily distortable.

Trees and branches in Banach spaces

An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree T of a certain type on a space X is presumed to have a branch with some property. It is shown that then X can be embedded into a space with an FDD (E i ) so that all normalized sequences in X which are almost a skipped blocking of (E i ) have that property. As an application of our work we prove that if X is a separable reflexive Banach space and for some 1 0, there exists a subspace of X having finite codimension which C 2 + e embeds into the l p sum of finite dimensional spaces.

Asymptotic properties of Banach spaces under renormings

It is shown that a separable Banach space

Banach spaces of bounded Szlenk index

X

A Banach space block finitely universal for monotone bases

can be given an equivalent norm

STRICTLY SINGULAR, NON-COMPACT OPERATORS EXIST ON THE SPACE OF GOWERS AND MAUREY

|\!|\!|\cdot |\!|\!|

An analytic solution to the Busemann-Petty problem

with the following properties:\quad If

On weakly null FDD'S in Banach spaces

(x_n)\subseteq X

Subsequential minimality in Gowers and Maurey spaces

is relatively weakly compact and