Before revision:  The result is an O(n^{d-1}) time algorithm for computing the simplicial depth of a point w in a set H of n d-dimensional points, improving on the previously best bound of O(n^d\log n).
After revision:  The result is an O(n^{d-1}) time algorithm for computing the simplicial depth of a point w in a set H of n points, improving on the previously best bound of O(n^d\log n).