  We introduce a new variant of quantitative Helly-type theorems: the minimal
\emph{"homothetic distance"} of the intersection of a family of convex sets to
the intersection of a subfamily of a fixed size. As an application, we
establish the following quantitative Helly-type result for the \emph{diameter}.
If $K$ is the intersection of finitely many convex bodies in $\mathbb{R}^d$,
then one can select $2d$ of these bodies whose intersection is of diameter at
most $(2d)^3\mathrm{diam}(K)$. The best previously known estimate, due to
Brazitikos, is $c d^{11/2}$. Moreover, we confirm that the multiplicative
factor $c d^{1/2}$ conjectured by B\'ar\'any, Katchalski and Pach cannot be
improved.
