Estimates for the minimal width of polytopes inscribed in convex bodies

The paper deals with the problem of approximating point sets by n-point subsets with respect to the minimal width w. Let, in particular, ℋ^d denote the family of all convex bodies in Euclidean d-space, let A ⊂ ℋ^d and let n be an integer greater than d. Then we ask for the greatest number μ=Λ_n(A) such that every A εA contains a polytope with n vertices which has minimal width at least μw(A). We give bounds for Λ_n(ℋ^d), for Λ_n(ℳ2133;^d), and for Λ_n(W^d), where ℳ2133;^d, W^d denote the families of centrally symmetric convex bodies and of bodies of constant width, respectively.