TY - GEN A1 - Averkov, Gennadiy A1 - Bröcker, Ludwig T1 - Minimal polynomial descriptions of polyhedra and special semialgebraic sets T2 - Advances in geometry N2 - 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) KW - Hörmander–Łojasiewicz’s Inequality KW - polyhedron KW - polynomial KW - polytope KW - semialge-braic set KW - stability index KW - Theorem of Bröcker and Scheiderer Y1 - 2012 UR - https://www.degruyter.com/view/j/advg.2012.12.issue-3/advgeom-2011-059/advgeom-2011-059.xml U6 - https://doi.org/10.1515/advgeom-2011-059 SN - 1615-715X SN - 1615-7168 VL - 12 IS - 3 SP - 447 EP - 459 ER - TY - GEN A1 - Averkov, Gennadiy A1 - Wagner, Christian T1 - Inequalities for the lattice width of lattice-free convex sets in the plane T2 - Beiträge zur Algebra und Geometrie N2 - 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. KW - Area Convex set KW - Covering minimum KW - Inhomogeneous minimum KW - Lattice-free body KW - Lattice width Y1 - 2012 UR - https://link.springer.com/article/10.1007%2Fs13366-011-0028-8 U6 - https://doi.org/10.1007/s13366-011-0028-8 SN - 2191-0383 SN - 0138-4821 VL - 53 IS - 1 SP - 1 EP - 23 ER - TY - GEN A1 - Averkov, Gennadiy T1 - A proof of Lovászs theorem on maximal lattice-free sets T2 - Beiträge zur Algebra und Geometrie N2 - 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. KW - Cutting plane KW - Lattice-free set KW - Lovász’s theorem Y1 - 2013 UR - https://link.springer.com/article/10.1007%2Fs13366-012-0092-8 U6 - https://doi.org/10.1007/s13366-012-0092-8 SN - 2191-0383 SN - 0138-4821 VL - 54 IS - 1 SP - 105 EP - 109 ER - TY - GEN A1 - Averkov, Gennadiy A1 - Conforti, Michelle A1 - Del Pia, Alberto A1 - Di Summa, Marco A1 - Faenza, Yuri T1 - On the convergence of the affine hull of the Chvátal-Gomory closures T2 - SIAM journal on discrete mathematics N2 - 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. KW - affine hul KW - Chvátal–Gomory closure KW - Chvátal rank KW - cutting plane KW - integral polyhe-dron Y1 - 2013 UR - https://epubs.siam.org/doi/10.1137/120898371 U6 - https://doi.org/10.1137/120898371 SN - 1095-7146 SN - 0895-4801 VL - 27 IS - 3 SP - 1492 EP - 1502 ER - TY - GEN A1 - Averkov, Gennadiy T1 - On maximal S-free sets and the helly number for the family of S-convex sets T2 - SIAM journal on discrete mathematics N2 - 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. KW - cutting plane KW - Doignon’s theorem KW - Helly’s theorem KW - Helly number KW - intersection cut KW - lattice-free set KW - S-convex set KW - S-free set Y1 - 2013 UR - https://epubs.siam.org/doi/10.1137/110850463 U6 - https://doi.org/10.1137/110850463 SN - 0895-4801 SN - 1095-7146 VL - 27 IS - 3 SP - 1610 EP - 1624 ER - TY - GEN A1 - Averkov, Gennadiy T1 - Constructive Proofs of some Positivstellensätze for Compact Semialgebraic Subsets of R d T2 - Journal of optimization theory and applications N2 - 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). KW - Polytope KW - Positivstellensatz KW - Preordering KW - Semiring KW - Quadratic module Y1 - 2013 UR - https://link.springer.com/article/10.1007%2Fs10957-012-0261-9 U6 - https://doi.org/10.1007/s10957-012-0261-9 SN - 1573-2878 SN - 0022-3239 VL - 158 IS - 2 SP - 410 EP - 418 ER - TY - CHAP A1 - Averkov, Gennadiy A1 - Basu, Amitabh ED - Lee, John ED - Vygen, Jens T1 - On the unique-lifting property T2 - Integer Programming and Combinatorial Optimization N2 - 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, Cornuéjols and Köppe [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. KW - Integer Point, Relative Interior, Tile Space, Unimodular Transformation, Nontrivial Generalization Y1 - 2014 SN - 978-3-319-07557-0 SN - 978-3-319-07556-3 U6 - https://doi.org/10.1007/978-3-319-07557-0_7 VL - 2014 SP - 76 EP - 87 PB - Springer ER - TY - GEN A1 - Averkov, Gennadiy A1 - Basu, Amitabh T1 - Lifting properties of maximal lattice-free polyhedra T2 - Mathematical Programming N2 - 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. Y1 - 2015 UR - https://link.springer.com/article/10.1007/s10107-015-0865-6 U6 - https://doi.org/10.1007/s10107-015-0865-6 SN - 0025-5610 VL - 154 IS - 1-2 SP - 81 EP - 111 ER - TY - GEN A1 - Averkov, Gennadiy A1 - Krümpelmann, Jan A1 - Nill, Benjamin T1 - Largest integral simplices with one interior integral point - solution of Hensley's conjecture and related results T2 - Advances in Mathematic N2 - 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. Y1 - 2015 UR - https://www.sciencedirect.com/science/article/pii/S0001870815000249?via%3Dihub#! U6 - https://doi.org/10.1016/j.aim.2014.12.035 SN - 0001-8708 VL - Volume 274 SP - 118 EP - 166 ER - TY - GEN A1 - Averkov, Gennadiy A1 - Bianchi, Gabriele T1 - Covariograms generated by valuations T2 - International Mathematics Research Notices N2 - 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. Y1 - 2015 UR - https://academic.oup.com/imrn/article/2015/19/9277/907311 U6 - https://doi.org/10.1093/imrn/rnu219 SN - 1687-0247 VL - Vol. 2015 IS - 19 SP - 9277 EP - 9329 ER - TY - GEN A1 - Averkov, Gennadiy A1 - González Merino, Bernardo A1 - Henze, Matthias A1 - Paschke, Ingo A1 - Weltge, Stefan T1 - Tight bounds on discrete quantitative Helly numbers T2 - arXiv.org : (math) N2 - 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. KW - Metric Geometry KW - Optimization and Control Y1 - 2016 UR - https://arxiv.org/abs/1602.07839 ER -