On the centrally symmetric ovals circumscribing invariant maximal quadrilaterals (Q2856411)

Let \(\mathcal C\) be a \(C^2\)-oval with center of symmetry. Fix a point \(A\) and consider other points \(B\), \(C\), \(D\) positioned clockwise on \(\mathcal C\). Say that the invariant maximal area (perimeter) assumption holds for \(\mathcal C\) if there exists a unique quadrilateral \(ABCD\) of maximal area (perimeter) and the resulting area (perimeter) is independent of \(A\). The author considers the questions: if any of these assumptions holds for an oval \(\mathcal C\) with center, does it have to be an ellipse?NEWLINENEWLINEIt is shown that when considered separately, each question has a negative answer because one quarter of any solution \(\mathcal C\) can be prescribed by imposing relatively mild restrictions. Applying to any eligible quarter a specific construction and the relevant invariant maximal property, one gets a neighboring quarter, and after that the completion of the oval follows from the required symmetry. More specifically, the ovals having the maximal perimeter property are precisely those ovals whose orthoptic curves are circles. The maximal area property holds for an initial oval selection provided that the product of the radii of curvature at the end-points equals the product of quarter oval semi-axes.NEWLINENEWLINEAnother result proved here is that an analytic oval which satisfies both invariant maximal properties is, as expected, an ellipse.