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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 eachset $A\subset X$ 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 ofBanach-Mazur metric, are presented. Two methods for estimating $\beta(X,m)$ areintroduced. 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 secondestimation 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 havingdiameters not greater than $\delta$ times the diameter of $K$, for certainclasses 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 thefirst method, and we proved that $\beta(X,8)<1$ whenever $X$ is athree-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. Theseresults and methods are closely related to the extension of Borsuk's problem infinite dimensional Banach spaces and to C. Zong's computer program for Borsuk'sconjecture.

Divide bounded sets into sets having smaller diameters