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Geometry of canonical self-similar tilings

We give several different geometric characterizations of the situation in which the parallel set $F_ε$ of a self-similar set $F$ can be described by the inner $ε$-parallel set $T_{-ε}$ of the associated canonical tiling $\mathcal T$, in the sense of \cite{SST}. For example, $F_ε=T_{-ε} \cup C_ε$ if and only if the boundary of the convex hull $C$ of $F$ is a subset of $F$, or if the boundary of $E$, the unbounded portion of the complement of $F$, is the boundary of a convex set. In the characterized situation, the tiling allows one to obtain a tube formula for $F$, i.e., an expression for the volume of $F_ε$ as a function of $ε$. On the way, we clarify some geometric properties of canonical tilings. Motivated by the search for tube formulas, we give a generalization of the tiling construction which applies to all self-affine sets $F$ having empty interior and satisfying the open set condition. We also characterize the relation between the parallel sets of $F$ and these tilings.