On the average area of a triangle inscribed in a convex figure

Let $K$ be a convex figure in the plane, and let $A, B, C$ be random points on its boundary given by a uniform distribution. In this paper, we prove that the maximum average area of triangle $ABC$ is obtained on the circle when the perimeter of $K$ is fixed. We also prove that the average area of the triangle is continuous in the Hausdorff metric as a functional of $K$.