Quantitative Helly-Type Theorems via Sparse Approximation

Abstract

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 $Q \subseteq - 2d^2\,Q'$. 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 $\mathcal {F}$ of convex bodies in $\mathbb {R}^d$ with intersection K, we may select at most 2d members of $\mathcal {F}$ such that their intersection has volume at most $(c d)^{3d /2} {{\,\textrm{vol}\,}}K$, and it has diameter at most $2d^2 {{\,\textrm{diam}\,}}K$, for some absolute constant $c>0$.