Given r > 1, we search for the convex body of minimal volume in E 3 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… (More)
We determine the minimal radius of n = 2, d or 2d congruent balls, which cover the d-dimensional crosspolytope.
For a given convex body K in R 3 with C 2 boundary, let P i n be the inscribed polytope of maximal volume with at most n vertices, and let P c (n) be the circumscribed polytope of minimal volume with at most n faces. We prove that typical faces of P i n are close to regular triangles in a suitable sense, and typical faces of P c (n) are close to regular… (More)