Limit behaviour of constant distance boundaries of Jordan curves

Feifei Qu; Xin Wei; Feifei Qu; Xin Wei

doi:10.3934/math.2022631

AIMS Mathematics

2022, Volume 7, Issue 6: 11311-11319. doi: 10.3934/math.2022631

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Research article

Limit behaviour of constant distance boundaries of Jordan curves

Feifei Qu ¹,
Xin Wei ^{2
,
,}

1.
School of Science, Tianjin University of Technology and Education, Tianjin, 300222, China
2.
School of Science, Xi'an Shiyou University, Xi'an, 710065, China

Received: 26 September 2021 Revised: 22 March 2022 Accepted: 25 March 2022 Published: 11 April 2022
MSC : 53A04, 57N40

For a Jordan curve $ \Gamma $ in the complex plane, its constant distance boundary $ \Gamma_ \lambda $ is an inflated version of $ \Gamma $. A flatness condition, $ (1/2, r_0) $-chordal property, guarantees that $ \Gamma_ \lambda $ is a Jordan curve when $ \lambda $ is not too large. We prove that $ \Gamma_ \lambda $ converges to $ \Gamma $, as $ \lambda $ approaching to $ 0 $, in the sense of Hausdorff distance if $ \Gamma $ has the $ (1/2, r_0) $-chordal property.
- constant distance boundary,
- Hausdorff distance,
- Jordan curve
Citation: Feifei Qu, Xin Wei. Limit behaviour of constant distance boundaries of Jordan curves[J]. AIMS Mathematics, 2022, 7(6): 11311-11319. doi: 10.3934/math.2022631

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Abstract

For a Jordan curve $ \Gamma $ in the complex plane, its constant distance boundary $ \Gamma_ \lambda $ is an inflated version of $ \Gamma $. A flatness condition, $ (1/2, r_0) $-chordal property, guarantees that $ \Gamma_ \lambda $ is a Jordan curve when $ \lambda $ is not too large. We prove that $ \Gamma_ \lambda $ converges to $ \Gamma $, as $ \lambda $ approaching to $ 0 $, in the sense of Hausdorff distance if $ \Gamma $ has the $ (1/2, r_0) $-chordal property.

References

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