The Intersection Property in the Family of Compact Convex Sets

We study compact convex sets having the following property: nonempty intersection of any family of translates of the set is a summand (in the sense of Minkowski) of that set. The intersection property was introduced by G. T. Sallee [J. Geometry 29 (1987)]. We call such sets Sallee sets. We prove that some sets other than polytopes and elipsoids, that is wedges, dull wedges (Theorem 3) and certain subsets of the Euclidean ball (Theorem 8), possess the intersection property. We also present the family of all three-dimensional polyhedral sets that have the intersection property (Theorem 5). The family coincides with the family of all three dimensional strongly monotypic polytopes, see the authors [Demonstratio Mathematica XLI(1) (2008) 165--169] and P. McMullen, R. Schneider and G. C. Shepard [Geometriae Dedicata 3 (1974) 99--129].