Optimal divisions of a convex body

For a convex body $C$ in ${\mathbb{R}}^d$ and a division of $C$ into convex subsets $C_1,\dots ,C_n$, we can consider $max\{F(C_1),\dots ,F(C_n)\}$ (respectively, $min\{F(C_1),\dots ,F(C_n)\}$), where $F$ represents one of these classical geometric magnitudes: the diameter, the minimal width, or the inradius. In this work we study the divisions of $C$ minimizing (respectively, maximizing) the previous value, as well as other related questions.