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

TL;DR

This paper investigates the maximum relative diameter functional in multi-rotationally symmetric convex bodies, analyzing how different rotational symmetries influence this measure and characterizing cases where these measures coincide.

Contribution

It introduces a study of the maximum relative diameter functional in multi-rotationally symmetric convex bodies and characterizes sets where symmetry-based measures are equal.

Findings

Established relations among diameter values for different symmetries

Characterized sets with coinciding diameter measures across symmetries

Analyzed properties of standard k-partitions in symmetric convex bodies

Abstract

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 $k\in\{k_1,\ldots,k_n\}\subset\nn$, 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 $k_i\geq3$) 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.