Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Notions of maximality for integral lattice-free polyhedra: the case of
  dimension three   Lattice-free sets (convex subsets of $\mathbb{R}^d$ without interior integer
points) and their applications for cutting-plane methods in mixed-integer
optimization have been studied in recent literature. Notably, the family of all
integral lattice-free polyhedra which are not properly contained in another
integral lattice-free polyhedron has been of particular interest. We call these
polyhedra $\mathbb{Z}^d$-maximal.
  It is known that, for fixed $d$, the family $\mathbb{Z}^d$-maximal integral
lattice-free polyhedra is finite up to unimodular equivalence. In view of
possible applications in cutting-plane theory, one would like to have a
classification of this family. However, this turns out to be a challenging task
already for small dimensions.
  In contrast, the subfamily of all integral lattice-free polyhedra which are
not properly contained in any other lattice-free set, which we call
$\mathbb{R}^d$-maximal lattice-free polyhedra, allow a rather simple geometric
characterization. Hence, the question was raised for which dimensions the
notions of $\mathbb{Z}^d$-maximality and $\mathbb{R}^d$-maximality are
equivalent. This was known to be the case for dimensions one and two. On the
other hand, Nill and Ziegler (2011) showed that for dimension $d \ge 4$, there
exist polyhedra which are $\mathbb{Z}^d$-maximal but not
$\mathbb{R}^d$-maximal. In this article, we consider the remaining case $d = 3$
and prove that for integral polyhedra the notions of $\mathbb{R}^3$-maximality
and $\mathbb{Z}^3$-maximality are equivalent. As a consequence, the
classification of all $\mathbb{R}^3$-maximal integral polyhedra by Averkov,
Wagner and Weismantel (2011) contains all $\mathbb{Z}^3$-maximal integral
polyhedra.

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The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: funct-an
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The categories of this paper are: gr-qc
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The categories of this paper are: plasm-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-th
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The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: atom-ph
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The categories of this paper are: physics
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-ex