On a convex operator for finite sets

Let S be a finite set with n elements in a real linear space. Let JS be a set of n intervals in R. We introduce a convex operator co(S,JS) which generalizes the familiar concepts of the convex hull conv S and the affine hull aff S of S. We establish basic properties of this operator. It is proved 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 can also be obtained. For example, this operator assigns a regular dodecagon to the 4-element set consisting of the vertices and the orthocenter of an equilateral triangle. For JS which consists of bounded intervals, we give the upper bound for the number of vertices of the polytope co(S,JS).