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    • Graduate School - MU Theses and Dissertations (MU)
    • Theses and Dissertations (MU)
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    • 2008 Dissertations (MU)
    • 2008 MU dissertations - Freely available online
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    Applications of fourier analysis to intersection bodies

    Schlieper, Jared
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    [PDF] research.pdf (201.9Kb)
    Date
    2008
    Format
    Thesis
    Metadata
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    Abstract
    The concept of an intersection body is central for the dual Brunn-Minkowski theory and has played an important role in the solution of the Busemann-Petty problem. A more general concept of [kappa]-intersection bodies is related to the generalization of the Busemann-Petty problem. We are interested in comparing classes of [kappa]-intersection bodies. In the first chapter we present the result that was published in J. Schlieper, A note on [kappa]-intersection bodies, Proceedings American. Mathematical Society,135 (2007), 2081-2088. The result examines the conjecture that the classes of [kappa]-intersection bodies increase with [kappa]. In particular, the result constructs a 4- intersection body that is not a 2-intersection body. The second chapter is concerned with the geometry of spaces of Lorentz type. We define a 1-homogeneous functional based on Lorentz type norms. Consider the family of norms [nearest integer function]x[nearest integer function][pi][alpha] = [alpha]i₁xq₁+ [alpha]inxqn₁/q where [alpha] = ([alpha]₁, . . . , [alpha]n) with [alpha]₁ [pi], . . . ,[pi] [alpha] [less than]0 and [pi]([alpha]) is [alpha] permutation of the vector [alpha]. Define a 1-homogeneous functional based on this family of norms as follows ... We examine the geometric properties of the space (Rn, [kappa].[kappa]k). First, we determine the conditions when the star body (Rn, [kappa].[kappa]) is a [kappa]-intersection body. Second, we find the extremal sections of the star body (Rn, [kappa].[kappa]). Throughout this work we use the Fourier Analytic methods that were recently developed.
    URI
    https://hdl.handle.net/10355/5529
    https://doi.org/10.32469/10355/5529
    Degree
    Ph. D.
    Thesis Department
    Mathematics (MU)
    Collections
    • 2008 MU dissertations - Freely available online
    • Mathematics electronic theses and dissertations (MU)

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