### 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 |
---|---|

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

**CONTROLLING LIPSCHITZ FUNCTIONS.** / Kupavskii, Andrey; Pach, János; Tardos, G.

Research output: Contribution to journal › Article

*Mathematika*, vol. 64, no. 3, pp. 898-910. https://doi.org/10.1112/S0025579318000311

}

TY - JOUR

T1 - CONTROLLING LIPSCHITZ FUNCTIONS

AU - Kupavskii, Andrey

AU - Pach, János

AU - Tardos, G.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85064131940&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85064131940&partnerID=8YFLogxK

U2 - 10.1112/S0025579318000311

DO - 10.1112/S0025579318000311

M3 - Article

AN - SCOPUS:85064131940

VL - 64

SP - 898

EP - 910

JO - Mathematika

JF - Mathematika

SN - 0025-5793

IS - 3

ER -