Geodesic
On the structure of self-affine convex bodies
Sbornik. Mathematics, Tome 204 (2013) no. 8, pp. 1122-1130

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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. Bibliography: 10 titles.
Keywords: self-affine set, convex polyhedron.
Mots-clés : partition 
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     title = {On the structure of self-affine convex bodies},
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A. S. Voynov. On the structure of self-affine convex bodies. Sbornik. Mathematics, Tome 204 (2013) no. 8, pp. 1122-1130. http://geodesic.mathdoc.fr/item/SM_2013_204_8_a1/