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Positivity

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Comparing volumes by concurrent cross-sections of complex lines: a Busemann–Petty type problem

  • Eric L. Grinberg
Article

Abstract

We consider the problem of comparing the volumes of two star bodies in an even-dimensional Euclidean space \({\mathbb {R}}^{2n} = {\mathbb {C}}^n\) by comparing their cross sectional areas along complex lines (special 2-dimensional real planes) through the origin. Under mild symmetry conditions on one of the bodies a Busemann–Petty type theorem holds. Quaternionic and octonionic analogs also hold. The argument relies on integration in polar coordinates coupled with Jensen’s inequality. Along the way we provide a criterion that detects which centered bodies are circular. i.e., stabilized by multiplication by complex numbers of unit modulus. Our goal is to present a Busemann–Petty type result with a minimum of required background (in the spirit of L. K. Hua’s book on the classical domains) and, in addition, to suggest characterizations of classes of star bodies by means of integral geometric inequalities.

Keywords

Busemann–Petty problem Complex cross-sections Star body Circular domain Volume characterization 

Mathematics Subject Classification

52A20 52A38 52A40 45A60 

Notes

Acknowledgements

The author wishes to thank Susanna Dann, David Feldman, Daniel Klain, Erwin Lutwak, Mehmet Orhon, Larry Zalcman and others for helpful discussions and the referees for suggesting several improvements in the manuscript.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of Massachusetts, BostonBostonUSA

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