Subdivisions of rotationally symmetric planar convex bodies minimizing the maximum relative diameter

TL;DR

This paper investigates how to subdivide rotationally symmetric convex bodies to minimize the maximum relative diameter, identifying optimal subdivisions and characterizing the best configurations for different numbers of parts.

Contribution

It proves that standard equiangular inradius segments minimize the maximum relative diameter for k-partitions with k ≥ 3, and characterizes the optimal sets for all k ≥ 3.

Findings

Standard k-partitions are optimal for k ≥ 3.

Optimal sets are intersections of the circle with regular k-gons.

The result holds for k ≤ 6 in general subdivisions.

Abstract

In this work we study subdivisions of $k$-rotationally symmetric planar convex bodies that minimize the maximum relative diameter functional. For some particular subdivisions called $k$-partitions, consisting of $k$ curves meeting in an interior vertex, we prove that the so-called \emph{standard $k$-partition} (given by $k$ equiangular inradius segments) is minimizing for any $k\in\mathbb{N}$, $k\geq 3$. For general subdivisions, we show that the previous result only holds for $k\leq 6$. We also study the optimal set for this problem, obtaining that for each $k\in\mathbb{N}$, $k\geq 3$, it consists of the intersection of the unit circle with the corresponding regular $k$-gon of certain area. Finally, we also discuss the problem for planar convex sets and large values of $k$, and conjecture the optimal $k$-subdivision in this case.