Covering a Euclidean Line or Hyperplane by Dilations of its Discretization

Abstract

The aim of this study is to find an optimal covering of a Euclidean straight line by dilations of two of its discretizations. As a starting point, we will refer to Azriel Rosenfeld’s chord property of 1974. We determine the optimal coverings of a Euclidean straight line using the dilations with structuring element centered at each point of a Euclidean straight line’s discretization, generalizing the results of Chassery and Sivignon (??2013). Then we generalize these cases to higher dimensions and establish a framework where we can get an optimal way to cover a Euclidean hyperplane by a digital one using dilations.