The authors generalize the following 3 results called “Archimedian properties of parabolas”: For any two points \(A\), \(B\) on a parabola \(k\) (described by a quadratic polynomial function \(f\)), \(S\) shall denote the area bounded by \(k\) and the line segment \(A,B\). If \(T\) denotes the maximum of the area of the triangle \(ABC\) with the point \(C\) on the parabola-segment between \(A\) and \(B\), we have \(S = {4\over 3} T\). The position of the corresponding vertex \(C\) with respect to \(A,B\) is the subject of the second property. The third Archimedian property offers a formula for the size \(S\) in terms of the coordinates of \(A\) and \(B\).
Some generalisations of the matter consist of the following new characterisations: If a convex function \(f\) of the class \(C^3\) satisfies a condition similar to the first Archimedian property, \(f\) has to be a polynomial function of degree 2 or less. Similarly, a statement analogous to the second Archimedian property characterizes quadratic polynomials, as well. The last Archimedian property characterizes \(f\) as a polynomial of degree 3 or less.