Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Determination of Boolean models by mean values of mixed volumes   In Weil (2001) formulas were proved for stationary Boolean models $Z$ in
$\mathbb{R}^d$ with convex or polyconvex grains, which express the densities of
mixed volumes of $Z$ in terms of related mean values of the underlying Poisson
particle process $X$. These formulas were then used to show that in dimensions
2 and 3 the mean values of $Z$ determine the intensity $\gamma$ of $X$. For
$d=4$ a corresponding result was also stated, but the proof given was
incomplete, since in the formula for the mean Euler characteristic $\overline
V_0 (Z)$ a term $\overline V^{(0)}_{2,2}(X,X)$ was missing. This was pointed
out in Goodey and Weil (2002), where it was also explained that a new
decomposition result for mixed volumes and mixed translative functionals would
be needed to complete the proof.
  Such a general decomposition result is now available based on flag measures
of the convex bodies involved (Hug, Rataj and Weil (2013, 2017)). Here, we show
that such flag representations not only lead to a correct derivation of the
4-dimensional result but even yield a corresponding uniqueness theorem in all
dimensions. In the proof of the latter, we make use of Alesker's representation
theorem for translation invariant valuations. We also discuss which shape
information can be obtained in this way and comment on the situation in the
non-stationary case.

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The categories of this paper are: cond-mat
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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas
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The categories of this paper are: physics
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The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
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The categories of this paper are: eess
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph