Let S be a closed set in the plane and let \documentclass[12pt]{minimal}
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$\alpha >0$\end{document}. The following results hold for the \documentclass[12pt]{minimal}
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$\alpha $\end{document}-kernel of S, denoted \documentclass[12pt]{minimal}
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$ {\rm Ker}_\alpha S$\end{document}. ¶¶1)¶ When S is bounded by a simple closed and locally connected curve, then \documentclass[12pt]{minimal}
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$ {\rm Ker}_\alpha S $\end{document} has nonempty interior if and only if for some \documentclass[12pt]{minimal}
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$ \epsilon >0 $\end{document}, every 3 points of S see via \documentclass[12pt]{minimal}
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$ \alpha $\end{document}-paths in S some pair a,b with \documentclass[12pt]{minimal}
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$ {\rm dist}\,(a,b)\geq \epsilon $\end{document}. The number 3 is best possible. ¶¶2)¶ When S is simply connected, then \documentclass[12pt]{minimal}
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$ {\rm Ker}_\alpha S=\cap \{M:M $\end{document} a maximal \documentclass[12pt]{minimal}
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$ \alpha $\end{document}-convex subset of \documentclass[12pt]{minimal}
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$ S\}$\end{document}.
