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arXiv · 1510.04181

A characterization of injective subsets in R^n with maximum norm

Dominic Descombes
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Abstract

We characterize all (absolute) 1-Lipschitz retracts Q of R^n 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 : R^{n-1} -> R of 1-Lipschitz maps with L \leq U and the inequalities read L(x_1,...,x_{i-1},x_{i+1},...,x_n) \leq x_i \leq U(x_1,...,x_{i-1},x_{i+1},...,x_n) 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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Dominic Descombes. 2015-10-14. A characterization of injective subsets in R^n with maximum norm. https://arxiv.org/abs/1510.04181

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