TY - JOUR

T1 - Some properties of convex hulls of integer points contained in general convex sets

AU - Dey, Santanu S.

AU - Morán R., Diego A.

PY - 2013/10

Y1 - 2013/10

N2 - 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 7:223-225, 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.

AB - 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 7:223-225, 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.

KW - Closedness

KW - Convex hull

KW - Convex integer program

UR - http://www.scopus.com/inward/record.url?scp=84875931891&partnerID=8YFLogxK

U2 - 10.1007/s10107-012-0538-7

DO - 10.1007/s10107-012-0538-7

M3 - Article

AN - SCOPUS:84875931891

VL - 141

SP - 507

EP - 526

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1-2

ER -