Caratheodory's classic result says that if a point p lies in the convex hull of a set P⊂Rd, then it lies in the convex hull of a subset Q⊂P of size at most d+1. What happens if we want a subset Q of size k<d+1 such that p∈convQ? In general, this is impossible as convQ is too low dimensional. We offer some remedy: p is close, in an appropriate sense, to convQ for some subset Q of size k. Similar results hold for Tverberg's theorem as well. This is joint work with Nabil Mustafa.