On some extremal problems in discrete geometry and applications of extremal graphs

Introducing the notions of distant pairs of vertices and big subtriangles of a polygon, and using extremal graph theory—specifically, Tutan's graph—we establish upper bounds for: the sum of distances between all distant pairs of vertices in polygons with unit perimeter, and the sum of areas of all big subtriangles in convex polygons with unit area. We also formulate a conjecture on the upper bound of the sum of the areas of all subtriangles in a convex polygon.