  Brazitikos' results on quantititative Helly-type theorems (for the volume and
for the diameter) rely on the work of Srivastava on sparsification of John's
decompositions. We change this technique by a stronger recent result due to
Friedland and Youssef. This, together with an appropriate selection in the
accuracy of the approximation, allow us to obtain Helly-type versions which are
sensitive to the number of convex sets involved.
