  We study a family of self-affine tiles in $\mathbb{R}^d$ ($d\ge2$) 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.
