Before revision:  In this paper, we settle this problem with an asymptotically optimal bound of O(1/\varepsilon^{(d-1)/2})  . Our result is based on a new relationship between \varepsilon-width caps of a convex body and its polar  . Using this relationship, we are able to obtain a volume-sensitive bound on the number of approximating caps that are "essentially different."
After revision:  In this paper, we show that it is possible to construct a polytope with O(1/\varepsilon^{(d-1)/2}) . Our result is based on a new relationship between \varepsilon-width caps of a convex body and its polar . Using this relationship, we are able to obtain a volume-sensitive bound on the number of approximating caps that are "essentially different."