A class of self-affine tiles in ${\mathbb{R}}^d$ that are $d$-dimensional tame balls

We study a family of self-affine tiles in ${\mathbb{R}}^d$ ($d\ge 2$) with noncollinear digit sets, which naturally generalizes a class studied originally by Deng and Lau in ${\mathbb{R}}^2$ and its extension to \mathbb{R}^3} by the authors. By using Brouwer’s invariance of domain theorem, along with a tool which we call horizontal distance, we obtain necessary and sufficient conditions for the tiles to be $d$-dimensional tame balls. This answers positively the conjecture in an earlier paper by the authors stating that a member in a certain class of self-affine tiles is homeomorphic to a $d$-dimensional ball if and only if its interior is connected.