On figures with maximal area, lying in the interior of a triangle, rectangle and convex polygons with a unitary width (Q2799769)

The paper deals with the following unsolved problem: Find the plane figure of maximal area, lying in the interior of a convex figure with a unitary width. This problem is investigated for a triangle, convex quadrilateral and convex polygon. The following theorems are proved:NEWLINENEWLINETheorem: The figure which lies in the interior of a convex quadrilateral with the unit width, has the area which is equal to \(1/6\) of the area of a circle with unit radius. The area equals \(\frac{\pi}{6}\).NEWLINENEWLINETheorem: In an arbitrary closed convex polygon with the unit width we can place a figure which consists of 2/3 of the arc of the circle of radius 1/3 and two tangents which touch this circle at the ends of these arcs. The area is equal to \(\frac{2}{27}\pi + \frac{2}{9}\).NEWLINENEWLINEFor the entire collection see [Zbl 1298.53003].