Estimates for the minimal width of polytopes inscribed in convex bodies

Abstract

The paper deals with the problem of approximating point sets by n-point subsets with respect to the minimal width w. Let, in particular, ℋd denote the family of all convex bodies in Euclidean d-space, let A ⊂ ℋd and let n be an integer greater than d. Then we ask for the greatest number μ=Λn(A) such that every A εA contains a polytope with n vertices which has minimal width at least μw(A). We give bounds for Λn(ℋd), for Λn(ℳ2133;d), and for Λn(Wd), where ℳ2133;d, Wd denote the families of centrally symmetric convex bodies and of bodies of constant width, respectively. © 1989 Springer-Verlag New York Inc.