Geodesic
A~Relative Ryshkov Perfect Polyhedron As a~Generatrix of a~$W$-Tiling
Informatics and Automation, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 52-62.

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For an arbitrary positive definite quadratic form $f$ in $n$ variables ($n$-PQF) and any positive number $\rho$, the notion of $(f,\rho)$-perfect $(n+m)$-PQF is introduced. The problem of finding all such forms for any given $n$-PQF $f$ and $\rho>0$ is studied. Two representations of all $(f,\rho)$-perfect $(n+1)$-PQFs are obtained: one in the form of the vertices of a tiling of the Euclidean $n$-space (we call this tiling a $W$-tiling corresponding to the $n$-PQF $f$ and the number $\rho$), and the other in the form of the vertices of an $n$-dimensional polyhedral surface $\mu _f(\rho )$ (we call it a relative Ryshkov perfect polyhedron corresponding to the $n$-PQF $f$ and the number $\rho$). It is proved that the polyhedron $\mu _f(\rho )$ is a generatrix of the $W$-tiling corresponding to the $n$-PQF $f$ and the number $\rho$. 
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R. G. Barykinskii. A~Relative Ryshkov Perfect Polyhedron As a~Generatrix of a~$W$-Tiling. Informatics and Automation, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 52-62. http://geodesic.mathdoc.fr/item/TRSPY_2002_239_a2/

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