Prove that if one convex polygon lies inside another one, then the perimeter of the inner polygon does not exceed the perimeter of the outer polygon.
The sum of the lengths of projections of a convex polygon sides on any line is equal to double length of the projection of a polygon on this line. Therefore, the sum of the lengths of projections of the sides’ vectors to any line for inner polygon is not greater than for the outer. Consequently, the sum of the lengths of the sides’ vectors, i.e. perimeter, of the inner polygon is not greater than the outer.