Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An extension of Minkowski's theorem and its applications to questions
  about projections for measures   Minkowski's Theorem asserts that every centered measure on the sphere which
is not concentrated on a great subsphere is the surface area measure of some
convex body, and, moreover, the surface area measure determines a convex body
uniquely.
  In this manuscript we prove an extension of Minkowski's theorem. Consider a
measure $\mu$ on $\mathbb{R}^n$ with positive degree of concavity and positive
degree of homogeneity. We show that a surface area measure of a convex set $K$,
weighted with respect to $\mu$, determines a convex body uniquely up to
$\mu$-measure zero. We also establish an existence result under natural
conditions including symmetry.
  We apply this result to extend the solution to classical Shephard's problem,
which asks the following: if one convex body in $\mathbb{R}^n$ has larger
projections than another convex body in every direction, does it mean that the
volume of the first convex body is also greater? The answer to this question is
affirmative when $n\leq 2$ and negative when $n\geq 3.$
  In this paper we introduce a new notion which relates projections of convex
bodies to a given measure $\mu$, and is a direct generalization of the Lebesgue
area of a projection. Using this notion we state a generalization of the
Shephard problem to measures and prove that the answer is affirmative for
$n\leq 2$ and negative for $n\geq 3$ for measures which have a positive degree
of homogeneity and a positive degree of concavity. We also prove stability and
separation results, and establish useful corollaries. Finally, we describe two
types of uniqueness results which follow from the extension of Minkowski's
theorem.

17
The categories of this paper are: q-fin
33
The categories of this paper are: plasm-ph
18
The categories of this paper are: econ
19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
34
The categories of this paper are: chem-ph
20
The categories of this paper are: acc-phys
35
The categories of this paper are: mtrl-th
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph