Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: funct-an
10
The categories of this paper are: q-bio
19
The categories of this paper are: eess
1
The categories of this paper are: cs
23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
20
The categories of this paper are: acc-phys
14
The categories of this paper are: math-ph
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph