  For each positive integer $m$ and each real finite dimensional Banach space
$X$, we set $\beta(X,m)$ to be the infimum of $\delta\in (0,1]$ such that each
set $A\subset X$ having diameter $1$ can be represented as the union of $m$
subsets of $A$ whose diameters are at most $\delta$. Elementary properties of
$\beta(X,m)$, including its stability with respect to $X$ in the sense of
Banach-Mazur metric, are presented. Two methods for estimating $\beta(X,m)$ are
introduced. The first one estimates $\beta(X,m)$ using the knowledge of
$\beta(Y,m)$, where $Y$ is a Banach space sufficiently close to $X$. The second
estimation uses the information about $\beta_X(K,m)$, the infimum of
$\delta\in(0,1]$ such that $K\subset X$ is the union of $m$ subsets having
diameters not greater than $\delta$ times the diameter of $K$, for certain
classes of convex bodies $K$ in $X$. In particular, we show that
$\beta(l_p^3,8)\leq 0.925$ holds for each $p\in [1,+\infty]$ by applying the
first method, and we proved that $\beta(X,8)<1$ whenever $X$ is a
three-dimensional Banach space satisfying $\beta_X(B_X,8)<\frac{221}{328}$,
where $B_X$ is the unit ball of $X$, by applying the second method. These
results and methods are closely related to the extension of Borsuk's problem in
finite dimensional Banach spaces and to C. Zong's computer program for Borsuk's
conjecture.
