# Checkerboards, Lipschitz functions and uniform rectifiability.

• Volume: 13, Issue: 1, page 189-210
• ISSN: 0213-2230

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## Abstract

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In his recent lecture at the International Congress [S], Stephen Semmes stated the following conjecture for which we provide a proof.Theorem. Suppose Ω is a bounded open set in Rn with n &gt; 2, and suppose that B(0,1) ⊂ Ω, Hn-1(∂Ω) = M &lt; ∞ (depending on n and M) and a Lipschitz graph Γ (with constant L) such that Hn-1(Γ ∩ ∂Ω) ≥ ε.Here Hk denotes k-dimensional Hausdorff measure and B(0,1) the unit ball in Rn. By iterating our proof we obtain a slightly stronger result which allows us to cover most of the unit sphere Sn-1.

## How to cite

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Jones, Peter W., Katz, Nets Hawk, and Vargas, Ana. "Checkerboards, Lipschitz functions and uniform rectifiability.." Revista Matemática Iberoamericana 13.1 (1997): 189-210. <http://eudml.org/doc/39527>.

@article{Jones1997,
abstract = {In his recent lecture at the International Congress [S], Stephen Semmes stated the following conjecture for which we provide a proof.Theorem. Suppose Ω is a bounded open set in Rn with n &gt; 2, and suppose that B(0,1) ⊂ Ω, Hn-1(∂Ω) = M &lt; ∞ (depending on n and M) and a Lipschitz graph Γ (with constant L) such that Hn-1(Γ ∩ ∂Ω) ≥ ε.Here Hk denotes k-dimensional Hausdorff measure and B(0,1) the unit ball in Rn. By iterating our proof we obtain a slightly stronger result which allows us to cover most of the unit sphere Sn-1.},
author = {Jones, Peter W., Katz, Nets Hawk, Vargas, Ana},
journal = {Revista Matemática Iberoamericana},
keywords = {Espacios lineales topológicos; Función lipschitziana; Curvas alabeadas; Grafos; Rectificación; Lipschitz graph; Hausdorff measure; rectifiability; checkerboard theorem},
language = {eng},
number = {1},
pages = {189-210},
title = {Checkerboards, Lipschitz functions and uniform rectifiability.},
url = {http://eudml.org/doc/39527},
volume = {13},
year = {1997},
}

TY - JOUR
AU - Jones, Peter W.
AU - Katz, Nets Hawk
AU - Vargas, Ana
TI - Checkerboards, Lipschitz functions and uniform rectifiability.
JO - Revista Matemática Iberoamericana
PY - 1997
VL - 13
IS - 1
SP - 189
EP - 210
AB - In his recent lecture at the International Congress [S], Stephen Semmes stated the following conjecture for which we provide a proof.Theorem. Suppose Ω is a bounded open set in Rn with n &gt; 2, and suppose that B(0,1) ⊂ Ω, Hn-1(∂Ω) = M &lt; ∞ (depending on n and M) and a Lipschitz graph Γ (with constant L) such that Hn-1(Γ ∩ ∂Ω) ≥ ε.Here Hk denotes k-dimensional Hausdorff measure and B(0,1) the unit ball in Rn. By iterating our proof we obtain a slightly stronger result which allows us to cover most of the unit sphere Sn-1.
LA - eng
KW - Espacios lineales topológicos; Función lipschitziana; Curvas alabeadas; Grafos; Rectificación; Lipschitz graph; Hausdorff measure; rectifiability; checkerboard theorem
UR - http://eudml.org/doc/39527
ER -

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