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.