Discrete homotopic distance between Lipschitz maps - Hoseinzadeh, Elahe; Mirebrahimi, Hanieh; Torabi, Hamid; Babaee, Ameneh (MATHEMATICA SCANDINAVICA, 2025)
In this paper, we investigate a discrete version of the homotopic distance between two s-Lipschitz maps for s ≥ 0. This distance is defined by specifying a step length r, such that two maps are considered homotopic if r is sufficiently large. In spaces with a significant number of holes, where no continuous homotopy exist and the homotopic distance equals infinite, the discrete homotopic distance provides a meaningful classification by effectively ignoring smaller holes. We show that the discrete homotopic distance D_r generalizes key concepts such as the discrete Lusternik-Schnirelmann category cat_r and the discrete topological complexity TC_r. Furthermore, we prove that D_r is invariant under discrete homotopy relations. This approach offers a flexible framework for classifying s-Lipschitz maps, loops, and paths based on the choice of r.