Gennadiy Averkov

Lifting properties of maximal lattice-free polyhedra

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,xa02012) 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.xa0(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.

Representing simple d -dimensional polytopes by d polynomials

A polynomial representation of a convex d-polytope P is a finite set {p1(x), . . . , pn(x)} of polynomials over

On the Unique-Lifting Property

{mathbb {R}^d}

Approximation of Corner Polyhedra with Families of Intersection Cuts

such that

Three-Dimensional Polyhedra Can Be Described by Three Polynomial Inequalities

{P={x in mathbb {R}^d : p_i(x) ge 0 mbox{ for every }1 le i le n}}

Homometry and Direct-Sum Decompositions of Lattice-Convex Sets

. Let s(d, P) be the least possible n as above. It is conjectured that s(d, P)xa0=xa0d 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.