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

2017/01/04 by Eric L. Grinberg, Grinberg, Eric L.
Mathematics · #52A20 #52A38 #52A40 #FOS: Mathematics #Mathematics and Applications #Metric Geometry (math.MG) #Morphological variations and asymmetry #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1701.02237

openalex publication_date 2017/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the problem of comparing the volumes of two star bodies in an\neven-dimensional euclidean space mathbb R²ⁿ = mathbb Cⁿ by comparing\ntheir cross sectional areas along complex lines (special 2-dimensional real\nplanes) through the origin. Under mild symmetry conditions on one of the bodies\na Busemann-Petty type theorem holds. Quaternionic and Octonionic analogs also\nhold. The argument relies on integration in polar coordinates coupled with\nJensen's inequality. Along the way we provide a criterion that detects which\ncentered bodies are it circular. i.e., stabilized by multiplication by\ncomplex numbers of unit modulus. Our goal is to present a Busemann-Petty type\nresult with a minimum of required background and, in addition, to suggest\ncharacterizations of classes of star bodies by means of integral geometric\ninequalities.\n

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