Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let S be a finite set with m elements in a real linear space and let J"S be a set of m intervals in R.  introduce a convex operator co(S,J"S) which generalizes the familiar concepts of the convex hull, convS, and the affine hull, affS, of S.  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 J"S  give two different upper bounds for the number of vertices of the polytopes produced as co(S,J"S). Our motivation comes from a recent improvement of the ll-known Gauss-Lucas theorem. It turns out that a particular convex set co(S,J"S) plays a central role in this improvement.
['Branko urgus', 'Krzysztof Kołodziejczyk']
I am writing a research paper on a idea On a convex operator for finite sets
[{"from": "human", "value": "I am writing a research paper on a idea On a convex operator for finite sets"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let S be a finite set with m elements in a real linear space and let J\"S be a set of m intervals in R.  introduce a convex operator co(S,J\"S) which generalizes the familiar concepts of the convex hull, convS, and the affine hull, affS, of S.  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 J\"S  give two different upper bounds for the number of vertices of the polytopes produced as co(S,J\"S). Our motivation comes from a recent improvement of the ll-known Gauss-Lucas theorem. It turns out that a particular convex set co(S,J\"S) plays a central role in this improvement."}]