A Relative Ryshkov Perfect Polyhedron As a Generatrix of a $W$-Tiling

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 $\rho$.