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

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Dominic Descombes

Metric Geometry · Submitted 14 Oct 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We characterize all (absolute) 1-Lipschitz retracts Q of R^n with the maximum norm.

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arXiv ID1510.04181 (version 1)
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CitationDominic Descombes. "A characterization of injective subsets in R^n with maximum norm." arXiv:1510.04181 (2015). https://pith.science/paper/FSRBNTNL
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,195 input tokens.

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arXiv listing abstract v1. Results and the main claim are underlined; .

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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What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We characterize all (absolute) 1-Lipschitz retracts Q of R^n with the maximum norm. contribution 0.91 0.93 0.70 asserted 1.00
s1 Omitting two technical details, they coincide with the subsets written as the solution set of (at most) 2n inequalities like follows. result 0.82 0.78 0.12 asserted 1.00
s2 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. result 0.95 0.83 0.61 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original 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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  • The proof of this characterization is based on the proof by the first author of the characterization in the particular case where $I={1,…,n}$, that is in the particular case where $l_∞(I)$ corresponds to $^n$ endowed with the maximum norm cf. [ref] .

    Injective Subsets of $l_{\infty}(I)$ · Introduction · 0.98

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