Typical faces of extremal polytopes with respect to a thin three-dimensional shell

Given r > 1, we search for the convex body of minimal volume in E3 that contains a unit ball, and whose extreme points are of distance at least r from the centre of the unit ball. It is known that the extremal body is the regular octahedron and icosahedron for suitable values of r. In this paper we prove that if r is close to one then the typical faces of the extremal body are asymptotically regular triangles. In addition we prove the analogous statement for the extremal bodies with respect to the surface area and the mean width. 1 Notation and known results Let us introduce the notation used throughout the paper. The implied constant in O(·) is always some absolute constant. For any notions related to convexity in this paper, consult R. Schneider [13] or P.M. Gruber [9]. We write o to denote the origin in En, 〈·, ·〉 to denote the scalar product, and ‖·‖ to denote the corresponding Euclidean norm. In addition for non–collinear points u,v,w, the angle of the half lines vu and vw is ∠(u,v,w). Given a set X ⊂ En, the affine hull and the interior of X are denoted by affX and intX , respectively. If X is compact convex then we write ∂X to denote the relative boundary of X with respect to affX . Moreover let [X1, . . . ,Xk] stand for the convex hull of the objects X1, . . . ,Xk. ∗Supported by OTKA grants T 043556 and 033752 †Supported by OTKA grants T 042769, 043520 and 049301, and the EU Marie Curie TOK project DISCCONVGEO