Divide bounded sets into sets having smaller diameters

Abstract

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.