The modified complex Busemann-Petty problem on sections of convex bodies

The modified complex Busemann-Petty problem on sections of convex bodies The complex Busemann-Petty problem asks whether origin symmetric convex bodies in $$\mathbb{C}^n$$ 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]. Positivity Springer Journals

The modified complex Busemann-Petty problem on sections of convex bodies

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Copyright © 2009 by Birkhäuser Verlag Basel/Switzerland
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
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