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Projections of convex bodies and the fourier transform

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Abstract

The Fourier analytic approach to sections of convex bodies has recently been developed and has led to several results, including a complete analytic solution to the Busemann-Petty problem, characterizations of intersection bodies, extremal sections ofl p-balls. In this article, we extend this approach to projections of convex bodies and show that the projection counterparts of the results mentioned above can be proved using similar methods. In particular, we present a Fourier analytic proof of the recent result of Barthe and Naor on extremal projections ofl p-balls, and give a Fourier analytic solution to Shephard’s problem, originally solved by Petty and Schneider and asking whether symmetric convex bodies with smaller hyperplane projections necessarily have smaller volume. The proofs are based on a formula expressing the volume of hyperplane projections in terms of the Fourier transform of the curvature function.

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, 65211, Columbia, MO, USA

    Alexander Koldobsky

  2. Department of Mathematics, Kansas State University, 66506, Manhattan, KS, USA

    Dmitry Ryabogin

  3. Department of Mathematics, University of Missouri, 65211, Columbia, MO, USA

    Artem Zvavitch

Authors
  1. Alexander Koldobsky
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  2. Dmitry Ryabogin
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  3. Artem Zvavitch
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Correspondence to Alexander Koldobsky.

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Koldobsky, A., Ryabogin, D. & Zvavitch, A. Projections of convex bodies and the fourier transform. Isr. J. Math. 139, 361–380 (2004). https://doi.org/10.1007/BF02787557

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  • DOI: https://doi.org/10.1007/BF02787557

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