On a convex operator for finite sets (Q2381542)

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