The irreducible decomposition of an affine homogeneous convex domain (Q1107848)

The authors study convex domains \(\Omega\) in R n without affine lines. Such a domain is called irreducible when it is not affinely equivalent to a product of convex domains of lower dimension. The main theorem of the paper says that \(\Omega\) is affinely equivalent to a product of irreducible domains \(\Omega_ i\) and the decomposition is unique up to affine equivalence. The same result was proved by Vinberg in the case of convex cones. The correspondence given by Vinberg between convex domains without affine lines and a particular class of \({\mathbb{R}}\)-algebras called clans is used to reduce the main theorem to an algebraic one. Some other results are proved, in particular about the isometry group of the canonical metric on \(\Omega\).