Trisections of a 3-rotationally symmetric planar convex body minimizing the maximum relative diameter

TL;DR

This paper investigates the problem of dividing a 3-rotationally symmetric convex shape into three parts to minimize the largest relative diameter, establishing the optimal trisection method and identifying the minimal possible maximum diameter.

Contribution

It proves that the standard trisection is optimal for this problem and characterizes the set that achieves the minimal maximum relative diameter.

Findings

Standard trisection is optimal for minimizing maximum relative diameter.

The optimal set and the universal lower bound are explicitly determined.

The study advances understanding of symmetric convex body partitioning.

Abstract

In this work we study the fencing problem consisting of finnding a trisection of a 3-rotationally symmetric planar convex body which minimizes the maximum relative diameter. We prove that an optimal solution is given by the so-called standard trisection. We also determine the optimal set giving the minimum value for this functional and study the corresponding universal lower bound.