Quantitative Helly-Type Theorems via Sparse Approximation

We prove the following sparse approximation result for polytopes. Assume that Q is a polytope in John's position. Then there exist at most 2d vertices of Q whose convex hull Q' satisfies Q subset of -2d(2) Q'. As a consequence, we retrieve the best bound for the quantitative Helly-type result for the volume, achieved by Brazitikos, and improve on the strongest bound for the quantitative Helly-type theorem for the diameter, shown by Ivanov and Naszodi: We prove that given a finite family F of convex bodies in R-d with intersection K, we may select at most 2d members of F such that their intersection has volume at most (cd)(3d)(/2) vol K, and it has diameter at most 2d(2) diam K, for some absolute constant c > 0.