A Quantitative Helly-Type Theorem: Containment in a Homothet

We introduce a new variant of quantitative Helly-type theorems: the minimal homothetic distance intersection family convex sets to subfamily fixed size. As an application, we establish following result for diameter. If $K$ is finitely many bodies in $\mathbb{R}^d$, then one can select $2d$ these whose diameter at most $(2d)^3{diam}(K)$. The best previously known estimate, due Brazitikos [Bull. Hellenic Math. Soc., 62 (2018), pp. 19--25], $c d^{11/2}$. Moreover, confirm that multiplicative factor d^{1/2}$ conjectured by Bárány, Katchalski, and Pach [Proc. Amer. 86 (1982), 109--114] cannot be improved. bounds above follow from our key concerns sparse approximation polytope hull well-chosen subset its vertices: Assume $Q \subset {\mathbb R}^d$ centroid origin. Then there exist 2d vertices $Q$ $Q^{\prime \prime}$ satisfies - 8d^3 Q^{\prime \prime}.$