The maximum relative diameter for multi-rotationally symmetric planar convex bodies

In this work we study the maximum relative diameter functional $d_M$ in the class of multi-rotationally symmetric planar convex bodies. A given set $C$ of this class is $k$-rotationally symmetric for $k\in \{k_1,\dots ,k_n\}\subset \text{nn}$, and so it is natural to consider the standard $k_i$-partition $P_{k_i}$ associated to $C$ (which is a minimizing $k_i$-partition for $d_M$ when $k_i\ge 3$) and the corresponding value $d_M(P_{k_i})$. We establish the relation among these values, characterizing the particular sets for which all these values coincide.