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 ofAs 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 2d 2 diam K, for some absolute constant c > 0.

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