On a problem of A. Pleijel

Abstract

In 1955, Arne Pleijel proposed the following problem which remains unsolved to this day: Given a closed plane convex curve C and a point x(2) at a fixed distance )~ above the plane, as the point x(2) varies, characterize the point for which the conical surface with vertex x(2) and base C attains its minimum, and determine the limits as 2 ~ 0 and 2 ~ oo of this minimum point. The purpose of this paper is to solve the cases where 2 approach its extremities and in the course of the solution, we obtain an interesting characterization of the limit points, which we shall call the Pleijel points of C. A consequence is that the inner Pleijel point provides an upper bound for the isoperimetric defect of C. We also generalize the problem to higher dimensional spaces, and obtain the corresponding characterizations of the limiting points for convex surfaces.

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

“…In his paper "On a problem of A. Pleijel " B. N. Cheng [1] locates the limiting point as h → 0 and as h → ∞ implicitly via a special boundary parametrization and the radius-of-curvature function ρ of ∂K. Using variational arguments he shows that the unique point x * (h) approaches the origin of the polar coordinate parametrization of ∂K whose support function λ(φ) is characterized by Furthermore, he shows that these characterizations are sufficient to determine the limiting points, if the domain is strictly convex.…”

“…The results about the location of the optimal point in the limits h → ∞ and h → 0 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]. …”