Geometry of self-affine tiles I

Abstract

For a self-similar or self-affine tile in R n we study the following questions: 1) What is the boundary? 2) What is the convex hull? We show that the boundary is a graph directed self-affine fractal, and in the self-similar case we give an algorithm to compute its dimension. We give necessary and sufficient conditions for the convex hull to be a polytope, and we give a description of the Gauss map of the convex hull.