Before revision:  Configurations equivalent to wheel sets have already been used by Perles for counting the faces of high-dimensional polytopes with few vertices via the Gale dual. Based on that we can compute the number of facets of the convex hull of n=d+k points in general position in \mathbb{R}^d in time O(n^{\max\{\omega,k-2\}}) where \omega \approx 2.373, even though the asymptotic number of facets may be as large as n^k.
After revision:  Based on our result about simplicial depth, we can compute the number of facets of the convex hull of n=d+k points in general position in \mathbb{R}^d in time O(n^{\max\{\omega,k-2\}}) where \omega \approx 2.373, even though the asymptotic number of facets may be as large as n^k.