Isoperimetric and Isoparametric Problems

1. INTRODUCTION. Two incongruent solids with remarkable properties are shown in Figure 1. One is a slice of a solid hemispherical shell with inner radius r and outer radius R cut by a plane parallel to the equator and at distance h < r from the equator. The other is a cylindrical shell with the same radii and altitude h. The surface of each solid consists of four components: an upper circular ring, a lower circular ring, an outer lateral surface, and an inner lateral surface. The two solids share the following properties: