  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 $0.966$.
