The geometry of circumscribing polygons of minimal perimeter (Q1321692)

Let \(C\) be a closed, strictly convex curve in \(\mathbb{R}^ 2\) and \(C(n)\) the class of all \(n\)-sided polygons which circumscribe \(C\). The author establishes necessary geometric conditions for a polygon \(\prod \in C(n)\) to have minimal perimeter. As an example it is derived that for \(C\) being a circle the \(\prod^* \in C(n)\) of minimal perimeter is the regular \(n\)-gon. Next, the special cases of circumscribing triangles and quadrilaterals are investigated in great detail. Also polygons \(\prod \in C(n)\) of minimal area are considered and the following result of M. M. Day (1947) is obtained: If \(\prod \in C(n)\) has minimal area, then \(\prod = V_ 1V_ 2 \dots V_ n\), \(V_ i = i\)-th vertex of \(\prod\), touches \(C\) at the midpoints of the edges \(\overline {V_ i V_{i+1}}\), \(i = 1,2, \dots, n\).