Before revision:  In the mid-1970's, a result by Dudley showed that O(1/\varepsilon^{(d-1)/2}) facets suffice, and Bronshteyn and Ivanov presented a similar bound on the number of vertices . While both results match known worst-case lower bounds , obtaining a similar upper bound on the total combinatorial complexity has been open for over 40 years. Recently, we made a first step forward towards this objective, obtaining a suboptimal bound.
After revision:  In the mid-1970's, a result by Dudley showed that O(1/\varepsilon^{(d-1)/2}) facets or vertices are possible, respectively, but neither achieves both bounds simultaneously.