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We study a family of self-affine tiles in $\mathbb{R}^d$ ($d\ge2$) withnoncollinear digit sets, which naturally generalizes a class studied originallyby Deng and Lau in $\mathbb{R}^2$ and its extension to $\mathbb{R}^3}$ by theauthors. By using Brouwer's invariance of domain theorem, along with a toolwhich we call horizontal distance, we obtain necessary and sufficientconditions for the tiles to be $d$-dimensional tame balls. This answerspositively the conjecture in an earlier paper by the authors stating that amember in a certain class of self-affine tiles is homeomorphic to a$d$-dimensional ball if and only if its interior is connected.

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