This is an announcement for the paper "The Busemann-Petty problem for arbitrary measures" by Artem Zvavitch.

Abstract: The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections. We apply this formula to study properties of general measures most of which were known before only in the case of the standard Lebesgue measure. We solve an analog of the Busemann-Petty problem for the case of general measures. In addition, we show that there are measures, for which the answer to the generalized Busemann-Petty problem is affirmative in all dimensions. Finally, we apply the latter fact to prove a number of different inequalities concerning the volume of sections of convex symmetric bodies in $\R^n$ and solve a version of generalized Busemann-Petty problem for sections by k-dimensional subspaces.

Archive classification: Metric Geometry; Functional Analysis

Mathematics Subject Classification: 52A15, 52A21, 52A38

The source file(s), GBP_Zvavitch.tex: 44254 bytes, is(are) stored in gzipped form as 0406406.gz with size 12kb. The corresponding postcript file has gzipped size 65kb.

Submitted from: zvavitch@math.kent.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/math.MG/0406406

or

http://arXiv.org/abs/math.MG/0406406

or by email in unzipped form by transmitting an empty message with subject line

uget 0406406

or in gzipped form by using subject line

get 0406406

to: math@arXiv.org.