A~Relative Ryshkov Perfect Polyhedron As a~Generatrix of a~$W$-Tiling
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 52-62
Voir la notice de l'article provenant de la source Math-Net.Ru
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 $\rho$.
@article{TM_2002_239_a2,
author = {R. G. Barykinskii},
title = {A~Relative {Ryshkov} {Perfect} {Polyhedron} {As} {a~Generatrix} of a~$W${-Tiling}},
journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
pages = {52--62},
publisher = {mathdoc},
volume = {239},
year = {2002},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TM_2002_239_a2/}
}
R. G. Barykinskii. A~Relative Ryshkov Perfect Polyhedron As a~Generatrix of a~$W$-Tiling. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 52-62. http://geodesic.mathdoc.fr/item/TM_2002_239_a2/