On a convex operator for finite sets
Let S be a finite set with m  elements in a real linear space and let JS J S be a set of m  intervals in R R . We introduce a convex operator co(S,JS) co ( S , J 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 JS J S we give two different upper bounds for the number of vertices of the polytopes produced as co(S,JS) co ( S , J S ) . Our motivation comes from a recent improvement of the well-known GaussLucas theorem. It turns out that a particular convex set co(S,JS) co ( S , J S ) plays a central role in this improvement.
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