Coxeter called the generalization of a parallelepiped in higher dimensions a parallelotope.
Specifically in n-dimensional space it is called n-dimensional parallelotope, or simply n-parallelotope. Thus a parallelogram is a 2-parallelotope and a parallelepiped is a 3-parallelotope.
The diagonals of an n-parallelotope intersect at one point and are bisected by this point. Inversion in this point leaves the n-parallelotope unchanged. See also fixed points of isometry groups in Euclidean space.
The edges radiating from one vertex of a k-parallelotope form a k-frame of the vector space, and the parallelotope can be recovered from these vectors, by taking linear combinations of the vectors, with weights between 0 and 1.
The n-volume of an n-parallelotope embedded in where can be computed by means of the Gram determinant. Alternatively, the volume is the norm of the exterior product of the vectors:
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