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

2023/10/04 by Licht, Martin W.
#51F30 #54E35 #54F45 #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2310.02865

Abstract

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 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)/(2L). These Lipschitz estimates are optimal. We also address the Lipschitz analysis of ℓ^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.

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