From: Brenda Sue Cook (brenda@math.missouri.edu)
Date: Fri Apr 18 2008 - 15:40:54 CDT
Ph.D. DEFENSE
Department of Mathematics
Marisa Zymonopoulou
(University of Missouri)
Sections of complex convex bodies
The Fourier analytic approach to sections of convex bodies has been
developed recently, and the main idea is to express
different parameters of a body in terms of the Fourier transform and
then apply methods of Fourier analysis to solve geometric problems.
The original Fourier approach applies to convex bodies in $\R^n.$
This thesis is focused at extending this approach to the complex
case, where
origin symmetric complex convex bodies are the unit balls of norms in
$\C^n.$
If considered as convex
bodies in $\R^{2n}$ complex convex bodies acquire the property of
invariance with respect to certain rotations. This crucial
observation arises from the nature of the norm of the bodies. Also
complex hyperplanes correspond to only few
of $(2n-2)$-dimensional subspaces of $\R^{2n}.$ These facts motivated
the study of the complex analogs of certain results on sections of
real convex bodies.
We present several results on sections of complex convex bodies by
complex hyperplanes in $\C^n,$ including the complex Busemann-Petty
problem.
Dissertation Advisor: A. Koldobsky
Friday, April 25
12:00 p.m.
312 Math Sciences
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