### Abstract

Given any positive integers m and d, we say a sequence of points (x _{i} ) _{i∈I} in R ^{m} is Lipschitz-d-controlling if one can select suitable values yi (i ∈ I) such that for every Lipschitz function f :R ^{m} →R ^{d} there exists i with /f (x _{i} ) - y _{i} /<1. We conjecture that for every m = d, a sequence (x _{i} ) _{i∈I} ⊂ R ^{m} is d-controlling if and only if [Equation presented here] We prove that this condition is necessary and a slightly stronger one is already sufficient for the sequence to be d-controlling. We also prove the conjecture for m = 1.

Original language | English |
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Pages (from-to) | 898-910 |

Number of pages | 13 |

Journal | Mathematika |

Volume | 64 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 2018 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematika*,

*64*(3), 898-910. https://doi.org/10.1112/S0025579318000311