Abstract.We establish some quantitative versions of Helly's famous theorem on convex sets in Euclidean space. We prove, for instance, that if C is any finite family of convex sets in Rd, such that the intersection of any 2d members of C has volume at least 1, then the intersection of all members belonging to C is of volume > d~d . A similar theorem is true for diameter, instead of volume. A quantitative version of Steinitz' Theorem is also proved.

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