On a problem of A. Pleijel (Q1205434)

\textit{A. Pleijel} [Math. Scand. 3, 306-307 (1955), Problem No. 16] proposed the following problem which remained unsolved until now [see also \textit{H. Croft}, \textit{K. J. Falconer}, and \textit{R. K. Guy}: Unsolved problems in geometry. Springer-Verlag (1991; Zbl 0748.52001), Problem A31]: Given a closed plane convex curve \(c\) and a point \(x(\lambda )\) at a finite distance \(\lambda \) above the plane. As the point \(x(\lambda )\) varies, characterize the point for which the area of the conical surface with this vertex and base \(c\) attains its minimum. In particular determine the limits of the minimum point as \(\lambda \to \infty\) and \(\lambda \to 0\). These points (Pleijel points) are determined by the author in the latter case. Also relations of these points to the isoperimetric defect of \(c\) and generalizations to convex hypersurfaces are shown in this paper.