Abstract:
For a bounded set $X$ with diameter $d_{C}(X)$ in a finite-dimensional normed space with an origin-symmetric convex body $C$ as the unit ball, the Borsuk number of $X$, denoted by $a_{C}(X)$, is the smallest integer $k$ such that $X$ can be represented as a union of $k$ sets, the diameter of each of which is strictly less than $d_{C}(X)$. In this paper, we solve the problem of finding the upper bound for the Borsuk number for any bounded set $X$ in the special Minkowski spaces $(\mathbb{R}^{n},\ell_{p})$. We have $a_{C}(X)\leq 2^{n}$ in $(\mathbb{R}^{n},\ell_{p})$, for all $p$ satisfying $1/(\log_{n}(n+1)-1)< p \leq + \infty$ and $2\leq n$, $n\in\mathbb{N}^{+}$. If $n=2$, we have $a_{C}(X)\leq 4$ for all values of $p$; this is proved by a new approach.
Cited by
