A major problem in computer graphics, image processing, numerical analysis and other applied areas is constructing a relevant object discretization. In a recent work [1] we investigated an approach for constructing a connected discretization of a set A⊆ℝ n by taking the integer points within an offset of A of a certain radius, and determined the minimal value of the offset radius which guarantees connectedness of the discretization, provided that A is path-connected. In the present paper we prove that the same results hold when A is connected but not necessarily path-connected. We also demonstrate similar facts about Hausdorff discretization, thus generalizing a theorem from [18]. The proofs combine approaches and techniques from [1] and [18].
['Valentin E. Brimkov']
On Connectedness of Discretized Objects