@misc{AverkovBroecker, author = {Averkov, Gennadiy and Br{\"o}cker, Ludwig}, title = {Minimal polynomial descriptions of polyhedra and special semialgebraic sets}, series = {Advances in geometry}, volume = {12}, journal = {Advances in geometry}, number = {3}, issn = {1615-715X}, doi = {10.1515/advgeom-2011-059}, pages = {447 -- 459}, abstract = {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 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)}, language = {en} } @misc{AverkovWagner, author = {Averkov, Gennadiy and Wagner, Christian}, title = {Inequalities for the lattice width of lattice-free convex sets in the plane}, series = {Beitr{\"a}ge zur Algebra und Geometrie}, volume = {53}, journal = {Beitr{\"a}ge zur Algebra und Geometrie}, number = {1}, issn = {2191-0383}, doi = {10.1007/s13366-011-0028-8}, pages = {1 -- 23}, abstract = {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 Lov{\´a}sz (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 Lov{\´a}sz (Ann Math (2) 128(3):577-602, 1988) with a new proof.}, language = {en} } @misc{Averkov, author = {Averkov, Gennadiy}, title = {A proof of Lov{\´a}szs theorem on maximal lattice-free sets}, series = {Beitr{\"a}ge zur Algebra und Geometrie}, volume = {54}, journal = {Beitr{\"a}ge zur Algebra und Geometrie}, number = {1}, issn = {2191-0383}, doi = {10.1007/s13366-012-0092-8}, pages = {105 -- 109}, abstract = {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 Lov{\´a}sz 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 Lov{\´a}sz's result has been published in a paper of Basu, Conforti, Cornu{\´e}jols 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 Lov{\´a}sz is relevant in integer and mixed-integer optimization.}, language = {en} } @misc{AverkovConfortiDelPiaetal., author = {Averkov, Gennadiy and Conforti, Michelle and Del Pia, Alberto and Di Summa, Marco and Faenza, Yuri}, title = {On the convergence of the affine hull of the Chv{\´a}tal-Gomory closures}, series = {SIAM journal on discrete mathematics}, volume = {27}, journal = {SIAM journal on discrete mathematics}, number = {3}, issn = {1095-7146}, doi = {10.1137/120898371}, pages = {1492 -- 1502}, abstract = {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 Chv{\´a}tal--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.}, language = {en} } @misc{Averkov, author = {Averkov, Gennadiy}, title = {On maximal S-free sets and the helly number for the family of S-convex sets}, series = {SIAM journal on discrete mathematics}, volume = {27}, journal = {SIAM journal on discrete mathematics}, number = {3}, issn = {0895-4801}, doi = {10.1137/110850463}, pages = {1610 -- 1624}, abstract = {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 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.}, language = {en} } @misc{Averkov, author = {Averkov, Gennadiy}, title = {Constructive Proofs of some Positivstellens{\"a}tze for Compact Semialgebraic Subsets of R d}, series = {Journal of optimization theory and applications}, volume = {158}, journal = {Journal of optimization theory and applications}, number = {2}, issn = {1573-2878}, doi = {10.1007/s10957-012-0261-9}, pages = {410 -- 418}, abstract = {In a broad sense, positivstellens{\"a}tze are results about representations of polynomials, strictly positive on a given set. We give proofs of some known positivstellens{\"a}tze 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, W{\"o}rmann (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).}, language = {en} } @incollection{AverkovBasu, author = {Averkov, Gennadiy and Basu, Amitabh}, title = {On the unique-lifting property}, series = {Integer Programming and Combinatorial Optimization}, volume = {2014}, booktitle = {Integer Programming and Combinatorial Optimization}, editor = {Lee, John and Vygen, Jens}, publisher = {Springer}, isbn = {978-3-319-07557-0}, doi = {10.1007/978-3-319-07557-0_7}, pages = {76 -- 87}, abstract = {We study the uniqueness of minimal liftings of cut generating functions obtained from maximal lattice-free polytopes. We prove a basic invariance property of unique minimal liftings for general maximal lattice-free polytopes. This generalizes a previous result by Basu, Cornu{\´e}jols and K{\"o}ppe [3] for simplicial maximal lattice-free polytopes, thus completely settling this fundamental question about lifting. We also extend results from [3] for minimal liftings in maximal lattice-free simplices to more general polytopes. These nontrivial generalizations require the use of deep theorems from discrete geometry and geometry of numbers, such as the Venkov-Alexandrov-McMullen theorem on translative tilings, and McMullen's characterization of zonotopes.}, language = {en} } @misc{AverkovBasu, author = {Averkov, Gennadiy and Basu, Amitabh}, title = {Lifting properties of maximal lattice-free polyhedra}, series = {Mathematical Programming}, volume = {154}, journal = {Mathematical Programming}, number = {1-2}, issn = {0025-5610}, doi = {10.1007/s10107-015-0865-6}, pages = {81 -- 111}, abstract = {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.}, language = {en} } @misc{AverkovKruempelmannNill, author = {Averkov, Gennadiy and Kr{\"u}mpelmann, Jan and Nill, Benjamin}, title = {Largest integral simplices with one interior integral point - solution of Hensley's conjecture and related results}, series = {Advances in Mathematic}, volume = {Volume 274}, journal = {Advances in Mathematic}, issn = {0001-8708}, doi = {10.1016/j.aim.2014.12.035}, pages = {118 -- 166}, abstract = {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.}, language = {en} } @misc{AverkovBianchi, author = {Averkov, Gennadiy and Bianchi, Gabriele}, title = {Covariograms generated by valuations}, series = {International Mathematics Research Notices}, volume = {Vol. 2015}, journal = {International Mathematics Research Notices}, number = {19}, issn = {1687-0247}, doi = {10.1093/imrn/rnu219}, pages = {9277 -- 9329}, abstract = {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.}, language = {en} } @misc{AverkovGonzalezMerinoHenzeetal., author = {Averkov, Gennadiy and Gonz{\´a}lez Merino, Bernardo and Henze, Matthias and Paschke, Ingo and Weltge, Stefan}, title = {Tight bounds on discrete quantitative Helly numbers}, series = {arXiv.org : (math)}, journal = {arXiv.org : (math)}, pages = {19}, abstract = {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)}). 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 <= 4.}, language = {en} } @misc{AverkovKruempelmannWeltge, author = {Averkov, Gennadiy and Kr{\"u}mpelmann, Jan and Weltge, Stefan}, title = {Notions of maximality for integral lattice-free polyhedra - the case of dimension three}, series = {arXiv.org : (math)}, journal = {arXiv.org : (math)}, pages = {45}, abstract = {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. 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. 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.}, language = {en} } @misc{AverkovKaibelWeltge, author = {Averkov, Gennadiy and Kaibel, Volker and Weltge, Stefan}, title = {Maximum semidefinite and linear extension complexity of families of polytopes}, series = {arXiv.org : (math)}, journal = {arXiv.org : (math)}, pages = {11}, language = {en} } @misc{AverkovLangfeld, author = {Averkov, Gennadiy and Langfeld, Barbara}, title = {Homometry and direct-sum decompositions of lattice-convex sets}, series = {Discrete \& computational geometry : an international journal of mathematics and computer science}, volume = {56}, journal = {Discrete \& computational geometry : an international journal of mathematics and computer science}, number = {1}, issn = {1432-0444}, doi = {10.1007/s00454-016-9786-2}, pages = {216 -- 249}, abstract = {Two sets in Rd are called homometric if they have the same covariogram, where the covariogram of a finite subset K of Rd is the function associating to each u∈Rd the cardinality of K∩(K+u). Understanding the structure of homometric sets is important for a number of areas of mathematics and applications. If two sets are homometric but do not coincide up to translations and point reflections, we call them nontrivially homometric. We study nontrivially homometric pairs of lattice-convex sets, where a set K is called lattice-convex with respect to a lattice M⊆Rd if K is the intersection of M and a convex subset of Rd. This line of research was initiated in 2005 by Daurat, G{\´e}rard and Nivat and, independently, by Gardner, Gronchi and Zong. All pairs of nontrivially homometric lattice-convex sets that have been known so far can essentially be written as direct sums S⊕T and S⊕(-T), where T is lattice-convex, the underlying lattice M is the direct sum of T and some sublattice L, and S is a subset of L. We study pairs of nontrivially homometric lattice-convex sets assuming this particular form and establish a necessary and a sufficient condition for the lattice-convexity of S⊕T. This allows us to explicitly describe all nontrivially homometric pairs in dimension two, under the above assumption, and to construct examples of nontrivially homometric pairs of lattice-convex sets for each d≥3.}, language = {en} } @misc{HenkAverkov, author = {Henk, Martin and Averkov, Gennadiy}, title = {Three-Dimensional Polyhedra Can Be Described by Three Polynomial Inequalities}, series = {Discrete \& computational geometry}, volume = {42}, journal = {Discrete \& computational geometry}, number = {2}, issn = {1432-0444}, doi = {10.1007/s00454-009-9183-1}, pages = {166 -- 186}, abstract = {Bosse et al. conjectured that for every natural number d≥2 and every d-dimensional polytope P in ℝ d , there exist d polynomials p 1(x),…,p d (x) satisfying P={x∈ℝ d :p 1(x)≥0,…,p d (x)≥0}. We show that every three-dimensional polyhedron can be described by three polynomial inequalities, which confirms the conjecture for the case d=3 but also provides an analogous statement for the case of unbounded polyhedra. The proof of our result is constructive.}, language = {en} } @misc{Averkov, author = {Averkov, Gennadiy}, title = {On nearly equilateral simplices and nearly l 8 spaces}, series = {Canadian mathematical bulletin}, volume = {53}, journal = {Canadian mathematical bulletin}, number = {3}, issn = {1496-4287}, doi = {10.4153/CMB-2010-055-1}, pages = {394 -- 397}, language = {en} } @misc{AverkovHenk, author = {Averkov, Gennadiy and Henk, Martin}, title = {Representing simple d-dimensional polytopes by d polynomials}, series = {Mathematical programming / A}, volume = {126}, journal = {Mathematical programming / A}, number = {2}, issn = {1436-4646}, doi = {10.1007/s10107-009-0280-y}, pages = {203 -- 230}, abstract = {A polynomial representation of a convex d-polytope P is a finite set {p 1(x), . . . , p n (x)} of polynomials over Rd such that P={x∈Rd:pi(x)≥0 for every 1≤i≤n}. Let s(d, P) be the least possible n as above. It is conjectured that s(d, P) = d for all convex d-polytopes P. We confirm this conjecture for simple d-polytopes by providing an explicit construction of d polynomials that represent a given simple d-polytope P.}, language = {en} } @misc{AverkovBey, author = {Averkov, Gennadiy and Bey, Christian}, title = {Description of polygonal regions by polynomials of bounded degree}, series = {Monatshefte f{\"u}r Mathematik}, volume = {162}, journal = {Monatshefte f{\"u}r Mathematik}, number = {1}, issn = {1436-5081}, doi = {10.1007/s00605-010-0224-x}, pages = {19 -- 27}, abstract = {We show that every (possibly unbounded) convex polygon P in R2 with m edges can be represented by inequalities p 1 ≥ 0, . . ., p n ≥ 0, where the p i 's are products of at most k affine functions each vanishing on an edge of P and n = n(m, k) satisfies s(m,k)≤n(m,k)≤(1+εm)s(m,k) with s(m,k) ≔ max {m/k, log2 m} and εm→0 as m→∞. This choice of n is asymptotically best possible. An analogous result on representing the interior of P in the form p 1 > 0, . . ., p n > 0 is also given. For k ≤ m/log2 m these statements remain valid for representations with arbitrary polynomials of degree not exceeding k.}, language = {en} } @misc{AverkovWeismantel, author = {Averkov, Gennadiy and Weismantel, R.}, title = {Transversal numbers over subsets of linear spaces}, series = {Advances in geometry}, volume = {12}, journal = {Advances in geometry}, number = {1}, issn = {1615-7168}, doi = {10.1515/advgeom.2011.028}, pages = {19 -- 28}, language = {en} } @misc{Averkov, author = {Averkov, Gennadiy}, title = {On the size of lattice simplices with a single interior lattice point}, series = {SIAM journal on discrete mathematics}, volume = {26}, journal = {SIAM journal on discrete mathematics}, number = {2}, issn = {1095-7146}, doi = {10.1137/110829052}, pages = {515 -- 526}, abstract = {Let \$\mathcal{T}^d\$ be the set of all d-dimensional simplices T in \$\mathbb{R}^d\$ with integer vertices and a single integer point in the interior of T. It follows from a result of Hensley that \$\mathcal{T}^d\$ is finite up to affine transformations that preserve \$\mathbb{Z}^d\$. It is known that when d grows, the maximum volume of the simplices \$T \in \mathcal{T}^d\$ becomes extremely large. We improve and refine bounds on the size of \$T \in \mathcal{T}^d\$ (where by the size we mean the volume or the number of lattice points). It is shown that each \$T \in \mathcal{T}^d\$ can be decomposed into an ascending chain of faces \$G_1 \subseteq \cdots \subseteq G_d=T\$ such that for every \$i \in \{1,\ldots,d\}\$, \$G_i\$ is i-dimensional and the size of \$G_i\$ is bounded from above in terms of i and d. The bound on the size of \$G_i\$ is double exponential in i. The presented upper bounds are asymptotically tight on the log-log scale.}, language = {en} }