Macías Virgós, EnriqueMosquera Lois, DavidPereira Sáez, María José2023-01-242023-01-242022Macías-Virgós, E., Mosquera-Lois, D. & Pereira-Sáez, M.J. Homotopic Distance and Generalized Motion Planning. Mediterr. J. Math. 19, 258 (2022). https://doi.org/10.1007/s00009-022-02166-41660-5446http://hdl.handle.net/10347/29989We prove that the homotopic distance between two maps defined on a manifold is bounded above by the sum of their subspace distances on the critical submanifolds of any Morse–Bott function. This generalizes the Lusternik–Schnirelmann theorem (for Morse functions) and a similar result by Farber for the topological complexity. Analogously, we prove that, for analytic manifolds, the homotopic distance is bounded by the sum of the subspace distances on any submanifold and its cut locus. As an application, we show how navigation functions can be used to solve a generalized motion planning problemeng© 2022 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/Atribución 4.0 Internacionalhttp://creativecommons.org/licenses/by/4.0/Morse–Bott functionTopological complexityL–S categoryHomotopic distanceCut locusjournal article10.1007/s00009-022-02166-41660-5454open access