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Steklov Mathematical Institute Seminar
March 15, 2001, Moscow, Steklov Mathematical Institute of RAS, Conference Hall (8 Gubkina)

A cycle of work on the theory of regular decompositions of spaces

S. S. Ryshkov, M. I. Shtogrin, N. P. Dolbilin

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Abstract: A famous theorem of Voronoi states that for each primitive parallelohedron there is an affinely equivalent Dirichlet–Voronoi parallelohedron (existence theorem). Ryshkov proved that such a DirichletVoronoi parallelohedron is unique up to similarity (uniqueness theorem).
He also proved that for each type of $n$-dimensional parallelohedron there are a finite number of so-called root (basic) parallelohedra of dimension at most $n$ and arranged in $\mathbf E^n$ in such a way that each parallelohedron of the indicated type is representable up to an affine transformation as a Minkowski sum with non-negative coefficients of these root parallelohedra. From a slight refinement of this result it follows, for example, that up to an affine transformation each fourdimensional parallelohedron can be decomposed into such a sum of a regular 24-hedron and its edges (Ryshkov).
It was proved that for a space of constant curvature and arbitrary dimension and for any discrete group of motions of it having a compact fundamental domain there are only finitely many combinatorial types of regular Dirichlet–Voronoi decompositions (Shtogrin).
So-called polycycles, which have important applications, were investigated. A polycycle is defined to be a cellular decomposition of the disk that admits a continuous locally homeomorphic cellular map to a Platonic decomposition of the sphere, the Euclidean plane, or the Lobachevskii plane. A criterion was established for a given graph to be the edge skeleton of some polycycle (Shtogrin).
It was shown that a simply connected $d$-dimensional space of constant curvature has a regular decomposition into polyhedra congruent to a given convex polyhedron $P$ if and only if around $P$ one can construct from polyhedra congruent to it a so-called $(d-2)$-corona having a certain radius and satisfying two specific conditions (the condition of stability of the corona group and the condition of local compatibility) (Dolbilin).
It was shown that if a family of decompositions of a space (Euclidean or Lobachevskii) with a finite protoset and with a certain local rule is at most countable, then among these decompositions at least one is crystallographic. This theorem generalizes known results on uncountability of socalled aperiodic families (Dolbilin).

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