Polytopes of Minimal Volume with Respect to a Shell - Another Characterization ofthe Octahedron and the Icosahedron

Let us introduce the notation used throughout the paper. For any notions related to convexity in this paper, consult R. Schneider [8]. We write o to denote the origin in the Euclidean space En, and ‖ · ‖ to denote the corresponding Euclidean norm. Given a set X ⊂ En, the affine hull and the convex hull of X are denoted by affX and convX , respectively, moreover the interior of X is denoted by intX . For a compact convex X , let ∂X denote the “relative boundary” of X ; namely, the boundary of X with respect to the topology of affX . The unit ball centred at o is denoted by Bn, and the boundary of Bn is denoted by Sn−1. As usual we call a compact convex set C with non–empty interior a convex body, and write V (C) to denote its volume, and S(C) to denote its surface area. The two–dimensional Hausdorff measure of a measurable subsetC of the boundary of some convex body in E3 is called the area A(C) ofC. The following statement is the starting point of our investigation: