Quantitative Helly-type theorems via sparse approximation

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 Nasz\'odi: We prove that given a finite family $\mathcal{F}$ of convex bodies in $\mathbb{R}^d$ with intersection $K$, we may select at most $2 d$ members of $\mathcal{F}$ such that their intersection has volume at most $(c d)^{3d /2} \,\mathrm{vol}\, K$, and it has diameter at most $2 d^2 \,\mathrm{diam} \,K$, for some absolute constant $c>0$.