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

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 $10⩽k⩽17$ 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 $0.966$.

19 pages, 17 figures