Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A characterization of some mixed volumes via the Brunn-Minkowski
  inequality   We consider a functional $\mathcal F$ on the space of convex bodies in $\R^n$
defined as follows: ${\mathcal F}(K)$ is the integral over the unit sphere of a
fixed continuous functions $f$ with respect to the area measure of the convex
body $K$. We prove that if $\mathcal F$ satisfies an inequality of
Brunn--Minkowski type, then $f$ is the support function of a convex body, i.e.,
$\mathcal F$ is a mixed volume. As a consequence, we obtain a characterization
of translation invariant, continuous valuations which are homogeneous of degree
$n-1$ and satisfy a Brunn--Minkowski type inequality.

15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc
21
The categories of this paper are: adap-org
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The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat
19
The categories of this paper are: eess
32
The categories of this paper are: atom-ph
34
The categories of this paper are: chem-ph
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci