  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.
