EXTREMAL TRIANGULATIONS OF CONVEX POLYGONS

T. Andreescu and O. Mushkarov asked the following Malfatti-type problem: What is the maximum total area of k nonoverlapping triangles placed in a given circle? In connection with this unsolved problem, it is proved that if each triangle must be inscribed in a given convex disc K, then the union of the triangles in the maximal arrangement is a convex (k - 2)-gon that is also inscribed in K. Furthemore, the presented constructions demonstrate that the nonoverlapping condition is essential in the sense that for any pair (n, k) with 1 < k < n - 2 there exists a convex n-gon in which the family of k triangles with maximal total area contains overlapping triangles. We also consider the analogous problem in which the total area is minimized.