The visible part of plane self-similar sets

Given a compact subset $F$ of $\mathbb{R}^2$, the visible part $V_θF$ of $F$ from direction $θ$ is the set of $x$ in $F$ such that the half-line from $x$ in direction $θ$ intersects $F$ only at $x$. It is suggested that if $\dim_H F \geq 1$ then $\dim_H V_θF = 1$ for almost all $θ$, where $\dim_H$ denotes Hausdorff dimension. We confirm this when $F$ is a self-similar set satisfying the convex open set condition and such that the orthogonal projection of $F$ onto every line is an interval. In particular the underlying similarities may involve arbitrary rotations and $F$ need not be connected.

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