Quantitative Helly-type theorems

Imre B{ár{ány, Meir Katchalski, J{ános Pach
1982 Proceedings of the American Mathematical Society  
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.
doi:10.1090/s0002-9939-1982-0663877-x fatcat:bhvcw3azfnahlno2c4uurbsxwa