Geometry of canonical self-similar tilings

Abstract

We give several different geometric characterizations of the situation in which the parallel set $F_{\epsilon }$ of a self-similar set $F$ can be described by the inner $\epsilon$-parallel set $T_{-\epsilon }$ of the associated canonical tiling $\mathcal{T}$, in the sense of \cite{SST}. For example, $F_{\epsilon }=T_{-\epsilon }\cup C_{\epsilon }$ 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 $F_{\epsilon }$ as a function of $\epsilon$. 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.

20 pages, 6 figures