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Let $X \subset \mathbb{R}^2$ be a subset of the plane with Hausdorff dimension $1<dim_H(X)<2$.

For a subset $Y \subset \mathbb{R}^n$ we define $Y$ to be convex if for every $y_1,y_2 \in Y$ the segment $[y_1,y_2]$ is entirely contained in $Y$.

Does there exist such $X$ that is also convex?

I was searching through the web but I'm not able to find a reference or something related to this type of question. Maybe it's just a silly one for the experts in the field, but I don't see how to address the problem by myself. Thanks in advance.

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  • $\begingroup$ Note that any convex set $X\subseteq\mathbb{R}^n$ has non-empty interior when considered as a subset of the affine subspace $A_X$ of $\mathbb{R}^n$ generated by $X$ $\endgroup$
    – Saúl RM
    Feb 28, 2023 at 15:53
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    $\begingroup$ Following Saúl, it means you want the Hausdorff dimension of $X$'s boundary, not of $X$. $\endgroup$ Feb 28, 2023 at 15:57
  • $\begingroup$ Slightly related (on the Hausdorff dimension of a graph for a given Hölder continuity exponent) $\endgroup$
    – Gro-Tsen
    Feb 28, 2023 at 16:43

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