Geodesic
Volumes of $L$-simplexes of five-dimensional lattices
Matematičeskie zametki, Tome 13 (1973) no. 5, pp. 771-782.

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We investigate the question of nonbasic Simplexes of an $L$-subdivision of five-dimensional lattices. It is shown that, apart from the Simplexes of volume $2V$ ($V$ is the volume of the basic simplex), no other nonbasic $L$-simplexes exist in these lattices. In primitive lattices the $L$-simplexes of doubled volume abut the basic $L$-simplexes by 4-dimensional faces. 
@article{MZM_1973_13_5_a15,
     author = {E. P. Baranovskii},
     title = {Volumes of $L$-simplexes of five-dimensional lattices},
     journal = {Matemati\v{c}eskie zametki},
     pages = {771--782},
     publisher = {mathdoc},
     volume = {13},
     number = {5},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_13_5_a15/}
}
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E. P. Baranovskii. Volumes of $L$-simplexes of five-dimensional lattices. Matematičeskie zametki, Tome 13 (1973) no. 5, pp. 771-782. http://geodesic.mathdoc.fr/item/MZM_1973_13_5_a15/