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  <doc>
    <id>24297</id>
    <completedYear/>
    <publishedYear>2012</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>447</pageFirst>
    <pageLast>459</pageLast>
    <pageNumber/>
    <edition/>
    <issue>3</issue>
    <volume>12</volume>
    <type>articler</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2019-08-09</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Minimal polynomial descriptions of polyhedra and special semialgebraic sets</title>
    <abstract language="eng">We show that a d-dimensional polyhedron S in Rd can be represented by d-polynomial inequalities, that is, S = fx 2 Rd : p0(x)  0; : : : ; pd(x) 0g, where p0; : : : ; pd1 are appropriate polynomials. Furthermore, if an elementary closed semialgebraic set S is given by&#13;
polynomials q1; : : : ; qk and for each x 2 S at most s of these polynomials vanish in x, then S can be represented by s + 1 polynomials (and by s polynomials)</abstract>
    <parentTitle language="eng">Advances in geometry</parentTitle>
    <identifier type="doi">10.1515/advgeom-2011-059</identifier>
    <identifier type="url">https://www.degruyter.com/view/j/advg.2012.12.issue-3/advgeom-2011-059/advgeom-2011-059.xml</identifier>
    <identifier type="issn">1615-715X</identifier>
    <identifier type="issn">1615-7168</identifier>
    <enrichment key="BTU">nicht an der BTU erstellt / not created at BTU</enrichment>
    <enrichment key="opus.doi.autoCreate">false</enrichment>
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    <author>
      <firstName>Gennadiy</firstName>
      <lastName>Averkov</lastName>
    </author>
    <submitter>
      <firstName>Angelika</firstName>
      <lastName>Breu</lastName>
    </submitter>
    <author>
      <firstName>Ludwig</firstName>
      <lastName>Bröcker</lastName>
    </author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Hörmander–Łojasiewicz’s Inequality</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>polyhedron</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>polynomial</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>polytope</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>semialge-braic set</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>stability index</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Theorem of Bröcker and Scheiderer</value>
    </subject>
    <collection role="institutes" number="1302">FG Algorithmische Mathematik</collection>
  </doc>
  <doc>
    <id>24298</id>
    <completedYear/>
    <publishedYear>2012</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>1</pageFirst>
    <pageLast>23</pageLast>
    <pageNumber/>
    <edition/>
    <issue>1</issue>
    <volume>53</volume>
    <type>articler</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2019-08-09</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Inequalities for the lattice width of lattice-free convex sets in the plane</title>
    <abstract language="eng">A closed, convex set K in R2 with non-empty interior is called lattice-free if the interior of K is disjoint with Z2. In this paper we study the relation between the area and the lattice width of a planar lattice-free convex set in the general and centrally symmetric case. A correspondence between lattice width on the one hand and covering minima on the other, allows us to reformulate our results in terms of covering minima introduced by Kannan and Lovász (Ann Math (2) 128(3):577–602, 1988). We obtain a sharp upper bound for the area for any given value of the lattice width. The lattice-free convex sets satisfying the upper bound are characterized. Lower bounds are studied as well. Parts of our results are applied in Averkov et al. (Maximal lattice-free polyhedra: finiteness and an explicit description in dimension three, http://arxiv.org/abs/1010.1077, 2010) for cutting plane generation in mixed integer linear optimization, which was the original inducement for this paper. We further rectify a result of Kannan and Lovász (Ann Math (2) 128(3):577–602, 1988) with a new proof.</abstract>
    <parentTitle language="deu">Beiträge zur Algebra und Geometrie</parentTitle>
    <identifier type="doi">10.1007/s13366-011-0028-8</identifier>
    <identifier type="url">https://link.springer.com/article/10.1007%2Fs13366-011-0028-8</identifier>
    <identifier type="issn">2191-0383</identifier>
    <identifier type="issn">0138-4821</identifier>
    <enrichment key="BTU">nicht an der BTU erstellt / not created at BTU</enrichment>
    <enrichment key="opus.doi.autoCreate">false</enrichment>
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    <author>
      <firstName>Gennadiy</firstName>
      <lastName>Averkov</lastName>
    </author>
    <submitter>
      <firstName>Angelika</firstName>
      <lastName>Breu</lastName>
    </submitter>
    <author>
      <firstName>Christian</firstName>
      <lastName>Wagner</lastName>
    </author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Area Convex set</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Covering minimum</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Inhomogeneous minimum</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Lattice-free body</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Lattice width</value>
    </subject>
    <collection role="institutes" number="1302">FG Algorithmische Mathematik</collection>
  </doc>
  <doc>
    <id>24299</id>
    <completedYear/>
    <publishedYear>2013</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>105</pageFirst>
    <pageLast>109</pageLast>
    <pageNumber/>
    <edition/>
    <issue>1</issue>
    <volume>54</volume>
    <type>articler</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2019-08-09</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">A proof of Lovászs theorem on maximal lattice-free sets</title>
    <abstract language="eng">Let K be a maximal lattice-free set in Rd , that is, K is convex and closed subset of Rd , the interior of K does not contain points of Zd and K is inclusion-maximal with respect to the above properties. A result of Lovász asserts that if K is d-dimensional, then K is a polyhedron with at most 2 d facets, and the recession cone of K is a linear space spanned by vectors from Zd . A first complete proof of mentioned Lovász’s result has been published in a paper of Basu, Conforti, Cornuéjols and Zambelli (where the authors use Dirichlet’s approximation as a tool). The aim of this note is to give another proof of this result. Our proof relies on Minkowki’s first fundamental theorem from the geometry of numbers. We remark that the result of Lovász is relevant in integer and mixed-integer optimization.</abstract>
    <parentTitle language="deu">Beiträge zur Algebra und Geometrie</parentTitle>
    <identifier type="doi">10.1007/s13366-012-0092-8</identifier>
    <identifier type="url">https://link.springer.com/article/10.1007%2Fs13366-012-0092-8</identifier>
    <identifier type="issn">2191-0383</identifier>
    <identifier type="issn">0138-4821</identifier>
    <enrichment key="BTU">nicht an der BTU erstellt / not created at BTU</enrichment>
    <enrichment key="opus.doi.autoCreate">false</enrichment>
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    <author>
      <firstName>Gennadiy</firstName>
      <lastName>Averkov</lastName>
    </author>
    <submitter>
      <firstName>Angelika</firstName>
      <lastName>Breu</lastName>
    </submitter>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Cutting plane</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Lattice-free set</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Lovász’s theorem</value>
    </subject>
    <collection role="institutes" number="1302">FG Algorithmische Mathematik</collection>
  </doc>
  <doc>
    <id>24301</id>
    <completedYear/>
    <publishedYear>2013</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>1492</pageFirst>
    <pageLast>1502</pageLast>
    <pageNumber/>
    <edition/>
    <issue>3</issue>
    <volume>27</volume>
    <type>articler</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2019-08-09</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">On the convergence of the affine hull of the Chvátal-Gomory closures</title>
    <abstract language="eng">Given an integral polyhedron $P\subseteq\mathbb{R}^n$ and a rational polyhedron $Q\subseteq\mathbb{R}^n$ containing the same integer points as $P$, we investigate how many iterations of the Chvátal--Gomory closure operator have to be performed on $Q$ to obtain a polyhedron contained in the affine hull of $P$. We show that if $P$ contains an integer point in its relative interior, then such a number of iterations can be bounded by a function depending only on $n$. On the other hand, we prove that if $P$ is not full-dimensional and does not contain any integer point in its relative interior, then no finite bound on the number of iterations exists.</abstract>
    <parentTitle language="eng">SIAM journal on discrete mathematics</parentTitle>
    <identifier type="doi">10.1137/120898371</identifier>
    <identifier type="url">https://epubs.siam.org/doi/10.1137/120898371</identifier>
    <identifier type="issn">1095-7146</identifier>
    <identifier type="issn">0895-4801</identifier>
    <enrichment key="BTU">nicht an der BTU erstellt / not created at BTU</enrichment>
    <enrichment key="opus.doi.autoCreate">false</enrichment>
    <enrichment key="opus.urn.autoCreate">false</enrichment>
    <author>
      <firstName>Gennadiy</firstName>
      <lastName>Averkov</lastName>
    </author>
    <submitter>
      <firstName>Angelika</firstName>
      <lastName>Breu</lastName>
    </submitter>
    <author>
      <firstName>Michelle</firstName>
      <lastName>Conforti</lastName>
    </author>
    <author>
      <firstName>Alberto</firstName>
      <lastName>Del Pia</lastName>
    </author>
    <author>
      <firstName>Marco</firstName>
      <lastName>Di Summa</lastName>
    </author>
    <author>
      <firstName>Yuri</firstName>
      <lastName>Faenza</lastName>
    </author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>affine hul</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Chvátal–Gomory closure</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Chvátal rank</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>cutting plane</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>integral polyhe-dron</value>
    </subject>
    <collection role="institutes" number="1302">FG Algorithmische Mathematik</collection>
  </doc>
  <doc>
    <id>24302</id>
    <completedYear/>
    <publishedYear>2013</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>1610</pageFirst>
    <pageLast>1624</pageLast>
    <pageNumber/>
    <edition/>
    <issue>3</issue>
    <volume>27</volume>
    <type>articler</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2019-08-09</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">On maximal S-free sets and the helly number for the family of S-convex sets</title>
    <abstract language="eng">We study two combinatorial parameters, which we denote by f(S) and h(S), associated with an arbitrary set S ⊆ Rd, where d ∈ N. In the nondegenerate situation, f(S) is the largest possible number of facets of a d-dimensional polyhedron L such that the interior of L is disjoint with S and L is inclusion-maximal with respect to this property. The parameter h(S) is the Helly number&#13;
of the family of all sets that can be given as the intersection of S with a convex subset of Rd. We obtain the inequality f(S) ≤ h(S) for an arbitrary S, and the equality f(S) = h(S) for every discrete S. Furthermore, motivated by research in integer and mixed-integer optimization, we show that 2d is the sharp upper bound on f(S) in the case S = (Zd × Rn) ∩ C, where n ≥ 0 and C ⊆ Rd+n is convex. The presented material generalizes and unifies results of various authors, including the result h(Zd) = 2d of Doignon, the related result f(Zd) = 2d of Lov´asz, and the inequality f(Zd ∩ C) ≤ 2d, which has recently been proved for every convex set C ⊆ Rd by Mor´an and Dey.</abstract>
    <parentTitle language="eng">SIAM journal on discrete mathematics</parentTitle>
    <identifier type="doi">10.1137/110850463</identifier>
    <identifier type="url">https://epubs.siam.org/doi/10.1137/110850463</identifier>
    <identifier type="issn">0895-4801</identifier>
    <identifier type="issn">1095-7146</identifier>
    <enrichment key="BTU">nicht an der BTU erstellt / not created at BTU</enrichment>
    <author>
      <firstName>Gennadiy</firstName>
      <lastName>Averkov</lastName>
    </author>
    <submitter>
      <firstName>Angelika</firstName>
      <lastName>Breu</lastName>
    </submitter>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>cutting plane</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Doignon’s theorem</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Helly’s theorem</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Helly number</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>intersection cut</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>lattice-free set</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>S-convex set</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>S-free set</value>
    </subject>
    <collection role="institutes" number="1302">FG Algorithmische Mathematik</collection>
  </doc>
  <doc>
    <id>24303</id>
    <completedYear/>
    <publishedYear>2013</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>410</pageFirst>
    <pageLast>418</pageLast>
    <pageNumber/>
    <edition/>
    <issue>2</issue>
    <volume>158</volume>
    <type>articler</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2019-08-09</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Constructive Proofs of some Positivstellensätze for Compact Semialgebraic Subsets of R d</title>
    <abstract language="eng">In a broad sense, positivstellensätze are results about representations of polynomials, strictly positive on a given set. We give proofs of some known positivstellensätze for compact semialgebraic subsets of ℝ d , which are to a large extent constructive and elementary. The presented proofs extend and simplify arguments of Berr, Wörmann (Manuscripta Math. 104(2):135–143, 2001) and Schweighofer (J. Pure Appl. Algebra 166(3):307–319, 2002; SIAM J. Optim. 15(3):805–825, 2005).</abstract>
    <parentTitle language="eng">Journal of optimization theory and applications</parentTitle>
    <identifier type="doi">10.1007/s10957-012-0261-9</identifier>
    <identifier type="url">https://link.springer.com/article/10.1007%2Fs10957-012-0261-9</identifier>
    <identifier type="issn">1573-2878</identifier>
    <identifier type="issn">0022-3239</identifier>
    <enrichment key="BTU">nicht an der BTU erstellt / not created at BTU</enrichment>
    <enrichment key="opus.doi.autoCreate">false</enrichment>
    <enrichment key="opus.urn.autoCreate">false</enrichment>
    <author>
      <firstName>Gennadiy</firstName>
      <lastName>Averkov</lastName>
    </author>
    <submitter>
      <firstName>Angelika</firstName>
      <lastName>Breu</lastName>
    </submitter>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Polytope</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Positivstellensatz</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Preordering</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Semiring</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Quadratic module</value>
    </subject>
    <collection role="institutes" number="1302">FG Algorithmische Mathematik</collection>
  </doc>
  <doc>
    <id>24329</id>
    <completedYear/>
    <publishedYear>2015</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>81</pageFirst>
    <pageLast>111</pageLast>
    <pageNumber/>
    <edition/>
    <issue>1-2</issue>
    <volume>154</volume>
    <type>articler</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2019-08-28</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Lifting properties of maximal lattice-free polyhedra</title>
    <abstract language="eng">We study the uniqueness of minimal liftings of cut-generating functions obtained from maximal lattice-free polyhedra. We prove a basic invariance property of unique minimal liftings for general maximal lattice-free polyhedra. This generalizes a previous result by Basu et al. (Math Oper Res 37(2):346–355, 2012) for simplicial maximal lattice-free polytopes, thus completely settling this fundamental question about lifting for maximal lattice-free polyhedra. We further give a very general iterative construction to get maximal lattice-free polyhedra with the unique-lifting property in arbitrary dimensions. This single construction not only obtains all previously known polyhedra with the unique-lifting property, but goes further and vastly expands the known list of such polyhedra. Finally, we extend characterizations from Basu et al. (2012) about lifting with respect to maximal lattice-free simplices to more general polytopes. These nontrivial generalizations rely on a number of results from discrete geometry, including the Venkov-Alexandrov-McMullen theorem on translative tilings and characterizations of zonotopes in terms of central symmetry of their faces.</abstract>
    <parentTitle language="eng">Mathematical Programming</parentTitle>
    <identifier type="doi">10.1007/s10107-015-0865-6</identifier>
    <identifier type="url">https://link.springer.com/article/10.1007/s10107-015-0865-6</identifier>
    <identifier type="issn">0025-5610</identifier>
    <enrichment key="BTU">nicht an der BTU erstellt / not created at BTU</enrichment>
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    <author>
      <firstName>Gennadiy</firstName>
      <lastName>Averkov</lastName>
    </author>
    <submitter>
      <firstName>Angelika</firstName>
      <lastName>Breu</lastName>
    </submitter>
    <author>
      <firstName>Amitabh</firstName>
      <lastName>Basu</lastName>
    </author>
    <collection role="institutes" number="1302">FG Algorithmische Mathematik</collection>
  </doc>
  <doc>
    <id>24330</id>
    <completedYear/>
    <publishedYear>2015</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>118</pageFirst>
    <pageLast>166</pageLast>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume>Volume 274</volume>
    <type>articler</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2019-08-28</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Largest integral simplices with one interior integral point - solution of Hensley's conjecture and related results</title>
    <abstract language="deu">For each dimension d, d-dimensional integral simplices with exactly one interior integral point have bounded volume. This was first shown by Hensley. Explicit volume bounds were determined by Hensley, Lagarias and Ziegler, Pikhurko, and Averkov. In this paper we determine the exact upper volume bound for such simplices and characterize the volume-maximizing simplices. We also determine the sharp upper bound on the coefficient of asymmetry of an integral polytope with a single interior integral point. This result confirms a conjecture of Hensley from 1983. Moreover, for an integral simplex with precisely one interior integral point, we give bounds on the volumes of its faces, the barycentric coordinates of the interior integral point and its number of integral points. Furthermore, we prove a bound on the lattice diameter of integral polytopes with a fixed number of interior integral points. The presented results have applications in toric geometry and in integer optimization.</abstract>
    <parentTitle language="eng">Advances in Mathematic</parentTitle>
    <identifier type="doi">10.1016/j.aim.2014.12.035</identifier>
    <identifier type="url">https://www.sciencedirect.com/science/article/pii/S0001870815000249?via%3Dihub#!</identifier>
    <identifier type="issn">0001-8708</identifier>
    <enrichment key="BTU">nicht an der BTU erstellt / not created at BTU</enrichment>
    <enrichment key="opus.doi.autoCreate">false</enrichment>
    <enrichment key="opus.urn.autoCreate">false</enrichment>
    <author>
      <firstName>Gennadiy</firstName>
      <lastName>Averkov</lastName>
    </author>
    <submitter>
      <firstName>Angelika</firstName>
      <lastName>Breu</lastName>
    </submitter>
    <author>
      <firstName>Jan</firstName>
      <lastName>Krümpelmann</lastName>
    </author>
    <author>
      <firstName>Benjamin</firstName>
      <lastName>Nill</lastName>
    </author>
    <collection role="institutes" number="1302">FG Algorithmische Mathematik</collection>
  </doc>
  <doc>
    <id>24331</id>
    <completedYear/>
    <publishedYear>2015</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>9277</pageFirst>
    <pageLast>9329</pageLast>
    <pageNumber/>
    <edition/>
    <issue>19</issue>
    <volume>Vol. 2015</volume>
    <type>articler</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
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    <title language="eng">Covariograms generated by valuations</title>
    <abstract language="eng">Let ϕ be a real-valued valuation on the family of compact convex subsets of Rn and let K be a convex body in Rn⁠. We introduce the ϕ-covariogram gK,ϕ of K as the function associating to each x∈Rn the value ϕ(K∩(K+x))⁠. If ϕ is the volume, then gK,ϕ is the covariogram, extensively studied in various sources. When ϕ is a quermassintegral (e.g., surface area or mean width) gK,ϕ has been introduced by Nagel [26]. We study various properties of ϕ-covariograms, mostly in the case n=2 and under the assumption that ϕ is translation invariant, monotone, and even. We also consider the generalization of Matheron's covariogram problem to the case of ϕ-covariograms, that is, the problem of determining an unknown convex body K⁠, up to translations and point reflections, by the knowledge of gK,ϕ⁠. A positive solution to this problem is provided under different assumptions, including the case where K is a polygon and ϕ is either strictly monotone or ϕ is the width in a given direction. We prove that there are examples in every dimension n≥3 where K is determined by its covariogram but it is not determined by its width-covariogram. We also present some consequence of this study in stochastic geometry.</abstract>
    <parentTitle language="eng">International Mathematics Research Notices</parentTitle>
    <identifier type="doi">10.1093/imrn/rnu219</identifier>
    <identifier type="url">https://academic.oup.com/imrn/article/2015/19/9277/907311</identifier>
    <identifier type="issn">1687-0247</identifier>
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    <author>
      <firstName>Gennadiy</firstName>
      <lastName>Averkov</lastName>
    </author>
    <submitter>
      <firstName>Angelika</firstName>
      <lastName>Breu</lastName>
    </submitter>
    <author>
      <firstName>Gabriele</firstName>
      <lastName>Bianchi</lastName>
    </author>
    <collection role="institutes" number="1302">FG Algorithmische Mathematik</collection>
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  <doc>
    <id>24332</id>
    <completedYear/>
    <publishedYear>2016</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber>19</pageNumber>
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    <completedDate>2019-08-28</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Tight bounds on discrete quantitative Helly numbers</title>
    <abstract language="eng">Given a subset S of R^n, let c(S,k) be the smallest number t such that whenever finitely many convex sets have exactly k common points in S, there exist at most t of these sets that already have exactly k common points in S. For S = Z^n, this number was introduced by Aliev et al. [2014] who gave an explicit bound showing that c(Z^n,k) = O(k) holds for every fixed n. Recently, Chestnut et al. [2015] improved this to c(Z^n,k) = O(k (log log k)(log k)^{-1/3} ) and provided the lower bound c(Z^n,k) = Omega(k^{(n-1)/(n+1)}).&#13;
We provide a combinatorial description of c(S,k) in terms of polytopes with vertices in S and use it to improve the previously known bounds as follows: We strengthen the bound of Aliev et al. [2014] by a constant factor and extend it to general discrete sets S. We close the gap for Z^n by showing that c(Z^n,k) = Theta(k^{(n-1)/(n+1)}) holds for every fixed n. Finally, we determine the exact values of c(Z^n,k) for all k &lt;= 4.</abstract>
    <parentTitle language="eng">arXiv.org : (math)</parentTitle>
    <identifier type="url">https://arxiv.org/abs/1602.07839</identifier>
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    <author>
      <firstName>Gennadiy</firstName>
      <lastName>Averkov</lastName>
    </author>
    <submitter>
      <firstName>Angelika</firstName>
      <lastName>Breu</lastName>
    </submitter>
    <author>
      <firstName>Bernardo</firstName>
      <lastName>González Merino</lastName>
    </author>
    <author>
      <firstName>Matthias</firstName>
      <lastName>Henze</lastName>
    </author>
    <author>
      <firstName>Ingo</firstName>
      <lastName>Paschke</lastName>
    </author>
    <author>
      <firstName>Stefan</firstName>
      <lastName>Weltge</lastName>
    </author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Metric Geometry</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Optimization and Control</value>
    </subject>
    <collection role="institutes" number="1302">FG Algorithmische Mathematik</collection>
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  <doc>
    <id>24333</id>
    <completedYear/>
    <publishedYear>2015</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber>45</pageNumber>
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    <completedDate>2019-08-28</completedDate>
    <publishedDate>--</publishedDate>
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    <title language="eng">Notions of maximality for integral lattice-free polyhedra - the case of dimension three</title>
    <abstract language="eng">Lattice-free sets (convex subsets of Rd 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 Zd-maximal.&#13;
It is known that, for fixed d, the family Zd-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.&#13;
In contrast, the subfamily of all integral lattice-free polyhedra which are not properly contained in any other lattice-free set, which we call Rd-maximal lattice-free polyhedra, allow a rather simple geometric characterization. Hence, the question was raised for which dimensions the notions of Zd-maximality and Rd-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≥4, there exist polyhedra which are Zd-maximal but not Rd-maximal. In this article, we consider the remaining case d=3 and prove that for integral polyhedra the notions of R3-maximality and Z3-maximality are equivalent. As a consequence, the classification of all R3-maximal integral polyhedra by Averkov, Wagner and Weismantel (2011) contains all Z3-maximal integral polyhedra.</abstract>
    <parentTitle language="eng">arXiv.org : (math)</parentTitle>
    <identifier type="url">https://arxiv.org/abs/1509.05200</identifier>
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    <author>
      <firstName>Gennadiy</firstName>
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    <submitter>
      <firstName>Angelika</firstName>
      <lastName>Breu</lastName>
    </submitter>
    <author>
      <firstName>Jan</firstName>
      <lastName>Krümpelmann</lastName>
    </author>
    <author>
      <firstName>Stefan</firstName>
      <lastName>Weltge</lastName>
    </author>
    <collection role="institutes" number="1302">FG Algorithmische Mathematik</collection>
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