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 $r$.