On a convex operator for finite sets

Let S be a finite set with m   elements in a real linear space and let JSJS be a set of m   intervals in RR. We introduce a convex operator co(S,JS)co(S,JS) which generalizes the familiar concepts of the convex hull, convS, and the affine hull, affS, of S  . We prove that each homothet of convS that is contained in affS can be obtained using this operator. A variety of convex subsets of affS 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 JSJS we give two different upper bounds for the number of vertices of the polytopes produced as co(S,JS)co(S,JS). Our motivation comes from a recent improvement of the well-known Gauss–Lucas theorem. It turns out that a particular convex set co(S,JS)co(S,JS) plays a central role in this improvement.