Geometry of canonical self-similar tilings
Abstract
We give several different geometric characterizations of the situation in which the parallel set Fε of a self-similar set F can be described by the inner ε-parallel set T-ε of the associated canonical tiling T, in the sense of SST. For example, Fε=T-ε Cε 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 Fε 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.