On a convex operator for finite sets

Let S be a finite set with in elements in a real linear space and let I (S) be a set of in intervals in R. We introduce a convex operator co(S, I (S)) which generalizes the familiar concepts of the convex hull, conv S, and the affine hull, aff S, of S. We prove that each homothet of conv S that is contained in aff S can be obtained using this operator. A variety of convex subsets of aff S with interesting combinatorial properties can also be obtained. For example, this operator can assign a regular dodecagon to the 4-element set consisting of the vertices and the orthocenter of an equilateral triangle. For two types of families I S we give two different upper bounds for the number of vertices of the polytopes produced as co(S, I (S)). Our motivation comes from a recent improvement of the well-known Gauss-Lucas theorem. It turns out that a particular convex set co(S, I (S)) plays a central role in this improvement. (C) 2007 Elsevier B.V. All fights reserved.