Helly-type theorems for approximate covering

Let F \cup {U} be a collection of convex sets in R^d such that F covers U. We prove that if the elements of F and U have comparable size then a small subset of F suffices to cover most of the volume of U; we analyze how small this subset can be depending on the geometry of the elements of F, and show that smooth convex sets and axis parallel squares behave differently. We obtain similar results for surface-to-surface visibility amongst balls in 3 dimensions for a notion of volume related to form factor. For each of these situations, we give an algorithm that takes F and U as input and computes in time O(|F|*|h_e|)$ either a point in U not covered by F or a subset h_e covering U up to a measure e, with |h_e| meeting our combinatorial bounds.