A characterization of injective subsets in Rn with maximum norm
Abstract
We characterize all (absolute) 1-Lipschitz retracts Q of Rn with the maximum norm. Omitting two technical details, they coincide with the subsets written as the solution set of (at most) 2n inequalities like follows. For every coordinate i=1,...,n, there is a lower and an upper bound L,U : Rn-1 -> R of 1-Lipschitz maps with L ≤ U and the inequalities read L(x1,...,xi-1,xi+1,...,xn) ≤ xi ≤ U(x1,...,xi-1,xi+1,...,xn) These sets are also exactly the injective subsets; meaning those Q such that every 1-Lipschitz map A -> Q, defined on a subset A of a metric space B, possesses a 1-Lipschitz extension B -> Q.
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