@MISC{A_ona, author = {Branko Ćurgus A and Krzysztof Koãlodziejczyk B}, title = {On a convex operator for finite sets}, year = {} }

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Abstract

Let S be a finite set with m elements in a real linear space and let JS be a set of m intervals in R. We introduce a convex operator co(S,JS) which generalizes the familiar concepts of the convex hull, convS, and the affine hull, aff S, of S. We prove that each homothet of convS 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 we give two different upper bounds for the number of vertices of the polytopes produced as 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) plays a central role in this improvement.