Geodesic
Minkowski reduction region for positive quadratic forms in seven variables
Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 4, Tome 67 (1977), pp. 108-143

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We find the faces of the region of the Minkowski reduction for $n=7$. 
@article{ZNSL_1977_67_a4,
     author = {P. P. Tammela},
     title = {Minkowski reduction region for positive quadratic forms in seven variables},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {108--143},
     publisher = {mathdoc},
     volume = {67},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1977_67_a4/}
}
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P. P. Tammela. Minkowski reduction region for positive quadratic forms in seven variables. Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 4, Tome 67 (1977), pp. 108-143. http://geodesic.mathdoc.fr/item/ZNSL_1977_67_a4/