TITLE

# Some properties of convex hulls of integer points contained in general convex sets

AUTHOR(S)
Dey, Santanu S.; Morán R., Diego A.
PUB. DATE
October 2013
SOURCE
Mathematical Programming;Oct2013, Vol. 141 Issue 1/2, p507
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
In this paper, we study properties of general closed convex sets that determine the closedness and polyhedrality of the convex hull of integer points contained in it. We first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed. This leads to useful results for special classes of convex sets such as pointed cones, strictly convex sets, and sets containing integer points in their interior. We then present a sufficient condition for the convex hull of integer points in general convex sets to be a polyhedron. This result generalizes the well-known result due to Meyer (Math Program 7:223â€“225, 1974 ). Under a simple technical assumption, we show that these sufficient conditions are also necessary for the convex hull of integer points contained in general convex sets to be a polyhedron.
ACCESSION #
90169574

## Related Articles

• Large transversals to small families of unit disks. Bisztriczky, T.; Fodor, F.; Oliveros, D. // Acta Mathematica Hungarica;2005, Vol. 106 Issue 4, p285

It is proved that if the nonempty intersection of bounded closed convex setsAnBis contained in (A+

• On inclusion and summands of bounded closed convex sets. Grzybowski, J.; Urbański, R. // Acta Mathematica Hungarica;2005, Vol. 106 Issue 4, p293

• An outcome space approach for generalized convex multiplicative programs. Oliveira, R�bia M.; Ferreira, Paulo A. V. // Journal of Global Optimization;May2010, Vol. 47 Issue 1, p107

This paper addresses the problem of minimizing an arbitrary finite sum of products of two convex functions over a convex set. Nonconvex problems in this form constitute a class of generalized convex multiplicative problems. Convex analysis results allow to reformulate the problem as an...

• A theorem on martingale selection for relatively open convex set-valued random sequences. Rokhlin, D. // Mathematical Notes;Apr/May2007, Vol. 81 Issue 3/4, p543

For set-valued random sequences ( G n) with relatively open convex values G n( Ï‰), we prove a new test for the existence of a sequence ( x n) of selectors adapted to the filtration and admitting an equivalent martingale measure. The statement is formulated in terms of the supports of regular...

• Local Convexity on Smooth Manifolds. Rapcsák, T. // Journal of Optimization Theory & Applications;Oct2005, Vol. 127 Issue 1, p165

Some properties of the spaces of paths are studied in order to define and characterize the local convexity of sets belonging to smooth manifolds and the local convexity of functions defined on local convex sets of smooth manifolds.

• Optimality conditions in global optimization and their applications. Rubinov, A. M.; Wu, Z. Y. // Mathematical Programming;Aug2009, Vol. 120 Issue 1, p101

In this paper we derive necessary and sufficient conditions for some problems of global minimization. Our approach is based on methods of abstract convexity: we use a representation of an upper semicontinuous function as the lower envelope of a family of convex functions. We discuss applications...

• On the Constructive Solution of Convex Programming Problems in Separable Form. Tchemisova, T. V. // Journal of Mathematical Sciences;Mar2004, Vol. 120 Issue 1, p1016

We present a constructive approach to solving convex programming problems in separable form and new constructive methods for simultaneously solving pairs of primal and dual geometric programming problems. These methods are based on the new principle of accumulation of approximate functions,...

• Isomorphic Steiner symmetrization. Klartag, B.; Milman, V.D. // Inventiones Mathematicae;Sep2003, Vol. 153 Issue 3, p463

This paper proves that there exist 3n Steiner symmetrizations that transform any convex set K?Rn into an isomorphic Euclidean ball; i.e. if vol(K)=vol(Dn) where Dn is the standard Euclidean unit ball, then K can be transformed into a body K such that c1Dn?K?c2Dn, where c1,c2 are numerical...

• Quasi-convex Functions in Carnot Groups*. Mingbao Sun; Xiaoping Yang // Chinese Annals of Mathematics;Apr2007, Vol. 28 Issue 2, p235

In this paper, the authors introduce the concept of h-quasiconvex functions on Carnot groups G. It is shown that the notions of h-quasiconvex functions and h-convex sets are equivalent and the L âˆž estimates of first derivatives of h-quasiconvex functions are given. For a Carnot group G of...

Share