Working on a topic whose research was initiated by Bárány, Katchalski and Pach in 1982, we study quantitative Helly-type theorems for the volume and the diameter of convex sets. 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 - 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 Naszódi: 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 2 d^2 diam K, for some absolute constant c > 0.
This a joint work with Víctor Hugo Almendra-Hernández and Gergely Ambrus.
