Cones based on reflection symmetric convex polygons: Remarks on a problem by A. Pleijel

In this note we investigate a problem formulated by Pleijel in 1955. It asks for the cone over a convex plane domain K having minimal surface among all cones over K with the same given height h. For cones based on reflection symmetric polygonal K we analyze the behaviour, as h go to 0 and as h go to infinity, of the position of the apex for the minimizing cone and characterize the coordinate of the limit points by necessary conditions. Furthermore, the question whether there are convex domains such that the minimal cone does not change with h is discussed. The results about the location of the optimal point in the limits h go to 0 and h go to infinity presented here give the more or less explicit algebraic coordinates of the optimal apex. A complete (but implicit) characterization of this point was given by B. Cheng in [1].