On Lipschitz partitions of unity and the Assouad--Nagata dimension

We show that the standard partition of unity subordinate to an open cover of a metric space has Lipschitz constant $\max (1,M-1)\mathrm{/}\mathcal{L}$, where $\mathcal{L}$ is the Lebesgue number and $M$ is the multiplicity of the cover. If the metric space satisfies the approximate midpoint property, such as length spaces do, then the upper bound improves to $(M-1)\mathrm{/}(2\mathcal{L})$. These Lipschitz estimates are optimal. We also address the Lipschitz analysis of ${\mathrm{\ell }}^p$-generalizations 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.