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It is well known that the Hausdorff dimension of the Koch curve is $\log_34\approx1.26>1$. Thus by the property of Hausdorff measure, $\forall1<p<1.2$, the $p$-Hausdorff measure of the Koch curve is $\infty$. On the other hand, the Koch curve consists of countably many line segments, whose $p$-Hausdorff measure is $0$, leading to the conclusion $p$-Hausdorff measure of the Koch curve is $0$. It seems to be a contradiction. Am I missing something? I am confused by my thoughts.

Arctic Char
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Liyang Shao
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  • "The Koch curve consists of countably many line segments." This is false. See https://en.wikipedia.org/wiki/Koch_snowflake – GEdgar Mar 29 '23 at 01:19
  • You might find the answers at https://math.stackexchange.com/questions/3005409/koch-curve-from-cantor-sets-paradox helpful. – Eric Wofsey Mar 29 '23 at 01:45
  • Thank you all! Now I know where I am wrong in defining the Koch curve. – Liyang Shao Mar 29 '23 at 01:52

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