Publication
We show that the standard partition of unity subordinate to an open cover of a metric space has Lipschitz constant max(1, M - 1)/L, where Lis the Lebesgue number and M is the multiplicity of the cover. If the metric space satisfies the approximate midpoint property, as length spaces do, then the upper bound improves to (M - 1)/(2L). These Lipschitz estimates are optimal. We also address the Lipschitz analysis of p-generalizations pound of the standard partition of unity, their partial sums, and their categorical products. Lastly, we characterize metric spaces with Assouad-Nagata dimension n as exactly those metric spaces for which every Lebesgue cover admits an open refinement with multiplicity n + 1 while reducing the Lebesgue number by at most a constant factor. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).