Metric dimension of the intersections of a point set with hyperplanes

AbstractIn this paper we improve the construction of Goto (1993) to obtain the Main Theorem: Let n, m and k be arbitrary integers such that 0 ⩽ m ⩽ n − 1 ⩾ 1 and m ⩽ k ⩽ min{2m, n − 1}. Then there exists a point set Xm,kn in Euclidean n-space Rn such that 1.(i) μdimXm,kn = m and dim Xm,kn = k,2.(ii) μdim(Xm,kn ∩ H) = m for every hyperplane H in Rn, and3.(iii) if either k < n − 1 or k = n − 1 = m, then dim(Xm,kn ∩ H) = k for every hyperplane H in Rn.Here dim (respectively μdim) denotes covering (respectively metric) dimension, and by a hyperplane in Rn we mean an (n − 1)-dimensional affine subspace of Rn