A construction of a subspace in Euclidean space with designated values of dimension and metric dimension

Abstract:

For every integer $m,k$, and $n$ such that $0 \leqslant m \leqslant n - 1$ and $m \leqslant k \leqslant \min \{ 2m,n - 1\}$, we construct a subspace $S_{m,k}^n$ in Euclidean $n$space ${{\mathbf {R}}^n}$ satisfying the conditions that $\mu \dim S_{m,k}^n = m$ and $\dim S_{m,k}^n = k$, where $\mu \dim$ denotes the metric dimension.