Absolutely minimal Lipschitz extension of tree-valued mappings

We prove that every Lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal Lipschitz extension (AMLE). We relate these extensions to a stochastic game called Politics—a generalization of a game called Tug of War that has been used in Peres et al. (J Am Math Soc 22(1):167–210, 2009) to study real-valued AMLEs.

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