On the structure of self-affine convex bodies

We study the structure of convex bodies in $\mathbb R^d$ that can be represented as a union of their affine images with no common interior points. Such bodies are called self-affine. Vallet's conjecture on the structure of self-affine bodies was proved for $d = 2$ by Richter in 2011. In the present paper we disprove the conjecture for all $d \geqslant 3$ and derive a detailed description of self-affine bodies in $\mathbb R^3$. Also we consider the relation between properties of self-affine bodies and functional equations with a contraction of an argument.
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