Borsuk's problem

The Borsuk number of a bounded set $F$ is the smallest natural number $k$ such that $F$ can be represented as a union of $k$ sets, the diameter of each of which is less than $\operatorname{diam}F$. In this paper we solve the problem of finding the Borsuk number of any bounded set in an arbitrary two-dimensional normed space (the solution is given in terms of the enlargement of a set to a figure of constant width). We indicate spaces for which the solution of Borsuk's problem has the same form as in the Euclidean plane.