Metric dimension of the intersections of a point set with hyperplanes

✍ Scribed by Tatsuo Goto

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 519 KB
• Volume: 82
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(97)00053-9

✦ Synopsis

In 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 3 1 and m < k < min{2m, n -1). Then there exists a point set Xk,, in Euclidean n-space IR" such that (i) pdimX& = m and dimXk,, = k, (ii) pdim(X& n H) = m for every hyperplane H in R", and (iii) if either k < n -1 or k = n -1 = m, then dim(XJ& n H) = k for every hyperplane H in R".

Here dim (respectively pdim) denotes covering (respectively metric) dimension, and by a hyperplane in IR" we mean an (n -I)-dimensional affine subspace of R".