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

TL;DR

This paper provides a complete characterization of injective subsets in R^n with the maximum norm, showing they are exactly the solution sets of specific inequalities involving 1-Lipschitz maps.

Contribution

It introduces a precise description of all injective subsets in R^n under the maximum norm using inequalities derived from 1-Lipschitz functions, linking geometric and functional properties.

Findings

Injective subsets are characterized by inequalities involving 1-Lipschitz functions.

These subsets coincide with 1-Lipschitz retracts of R^n with the maximum norm.

The characterization simplifies understanding of Lipschitz extension properties in this setting.

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.