The modified complex Busemann-Petty problem on sections of convex bodies |
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Authors: | Marisa Zymonopoulou |
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Affiliation: | (1) Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106, USA |
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Abstract: | The complex Busemann-Petty problem asks whether origin symmetric convex bodies in with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if n ≤ 3 and negative if n ≥ 4. Since the answer is negative in most dimensions, it is natural to ask what conditions on the (n − 1)-dimensional volumes of the central sections of complex convex bodies with complex hyperplanes allow to compare the n-dimensional volumes. In this article we give necessary conditions on the section function in order to obtain an affirmative answer in all dimensions. The result is the complex analogue of [16]. |
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Keywords: | Convex bodies sections Fourier transform |
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