[Banach] Abstract of a paper by Artem Zvavitch

[Dale Alspach]

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 at math.kent.edu

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

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

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

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