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15.
CONSTANT DISTANCE BOUNDARIES OF THE $t$-QUASICIRCLE AND THE KOCH SNOWFLAKE CURVE*
Xin Wei, Zhi-Ying Wen
数学物理学报(英文版)
2023, 43 (3):
981-993.
DOI: 10.1007/s10473-023-0301-6
Let $\Gamma$ be a Jordan curve in the complex plane and let $\Gamma_\lambda$ be the constant distance boundary of $\Gamma$. Vellis and Wu \cite{VW} introduced the notion of a $(\zeta,r_0)$-chordal property which guarantees that, when $\lambda$ is not too large, $\Gamma_\lambda$ is a Jordan curve when $\zeta=1/2$ and $\Gamma_\lambda$ is a quasicircle when $0<\zeta<1/2$. We introduce the $(\zeta,r_0,t)$-chordal property, which generalizes the $(\zeta,r_0)$-chordal property, and we show that under the condition that $\Gamma$ is $(\zeta,r_0,\sqrt t)$-chordal with $0<\zeta < r_0^{1-\sqrt t}/2$, there exists $\varepsilon>0$ such that $\Gamma_\lambda$ is a $t$-quasicircle once $\Gamma_\lambda$ is a Jordan curve when $0<\zeta<\varepsilon$. In the last part of this paper, we provide an example: $\Gamma$ is a kind of Koch snowflake curve which does not have the $(\zeta,r_0)$-chordal property for any $0<\zeta\le 1/2$, however $\Gamma_\lambda$ is a Jordan curve when $\zeta$ is small enough. Meanwhile, $\Gamma$ has the $(\zeta,r_0,\sqrt t)$-chordal property with $0<\zeta < r_0^{1- \sqrt t}/2$ for any $t\in (0,1/4)$. As a corollary of our main theorem, $\Gamma_\lambda$ is a $t$-quasicircle for all $0<t<1/4$ when $\zeta$ is small enough. This means that our $(\zeta,r_0,t)$-chordal property is more general and applicable to more complicated curves.
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