Coverings of planar and three-dimensional sets with subsets of smaller diameter

TL;DR

This paper investigates the problem of covering planar and three-dimensional sets with subsets of smaller diameter, providing new bounds and algorithms for minimal diameters in such coverings.

Contribution

It introduces an algorithm for finding sub-optimal partitions and improves bounds on minimal diameters for coverings in 2D and 3D cases.

Findings

Improved upper and lower estimates for minimal diameters in planar sets.

An algorithm for sub-optimal partitioning of sets.

Partitioning of 3D sets into four subsets with diameter ≤ 0.966.

Abstract

Quantitative estimates related to the classical Borsuk problem of splitting set in Euclidean space into subsets of smaller diameter are considered. For a given $k$ there is a minimal diameter of subsets at which there exists a covering with $k$ subsets of any planar set of unit diameter. In order to find an upper estimate of the minimal diameter we propose an algorithm for finding sub-optimal partitions. In the cases $10 \leqslant k \leqslant 17$ some upper and lower estimates of the minimal diameter are improved. Another result is that any set $M \subset \mathbb{R}^3$ of a unit diameter can be partitioned into four subsets of a diameter not greater than $0.966$.