Quantitative Helly-type theorems

Abstract:

We establish some quantitative versions of Helly’s famous theorem on convex sets in Euclidean space. We prove, for instance, that if $\mathcal {C}$ is any finite family of convex sets in ${{\mathbf {R}}^d}$, such that the intersection of any $2d$ members of $\mathcal {C}$ has volume at least 1, then the intersection of all members belonging to $\mathcal {C}$ is of volume $\geqslant {d^{ - {d^2}}}$. A similar theorem is true for diameter, instead of volume. A quantitative version of Steinitz’ Theorem is also proved.