A1 Journal article (refereed)
Sharp estimate on the inner distance in planar domains (2020)


Lučić, Danka; Pasqualetto, Enrico; Rajala, Tapio (2020). Sharp estimate on the inner distance in planar domains. Arkiv för Matematik, 58 (1), 133-159. DOI: 10.4310/ARKIV.2020.v58.n1.a9


JYU authors or editors


Publication details

All authors or editors: Lučić, Danka; Pasqualetto, Enrico; Rajala, Tapio

Journal or series: Arkiv för Matematik

ISSN: 0004-2080

eISSN: 1871-2487

Publication year: 2020

Volume: 58

Issue number: 1

Pages range: 133-159

Publisher: International Press

Publication country: Sweden

Publication language: English

DOI: http://doi.org/10.4310/ARKIV.2020.v58.n1.a9

Open Access: Publication published in an open access channel

Web address of parallel published publication (pre-print): https://arxiv.org/abs/1905.07988


Abstract

We show that the inner distance inside a bounded planar domain is at most the one-dimensional Hausdorff measure of the boundary of the domain. We prove this sharp result by establishing an improved Painlevé length estimate for connected sets and by using the metric removability of totally disconnected sets, proven by Kalmykov, Kovalev, and Rajala. We also give a totally disconnected example showing that for general sets the Painlevé length bound ϰ(E)≤πH1(E) is sharp.


Keywords: mathematical sciences; metric spaces

Free keywords: inner distance; Painlevé length; accessible points


Contributing organizations


Related projects

Centre of Excellence in Analysis and Dynamics Research
Koskela, Pekka
Academy of Finland
01/01/2017-31/12/2019
Geometric Aspects of Sobolev Space Theory
Rajala, Tapio
Academy of Finland
01/09/2018-31/08/2022


Ministry reporting: Yes

Reporting Year: 2020

Preliminary JUFO rating: 2


Last updated on 2020-09-07 at 23:12