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

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 {\it 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 and, in addition, to suggest characterizations of classes of star bodies by means of integral geometric inequalities.