It could be any shape, although the volume equals that of the parallelotope.
2.
Inversion in this point leaves the " n "-parallelotope unchanged.
3.
Geometrically, the Gram determinant is the square of the volume of the parallelotope formed by the vectors.
4.
More generally a parallelotope, or " voronoi parallelotope ", has parallel and congruent opposite facets.
5.
More generally a parallelotope, or " voronoi parallelotope ", has parallel and congruent opposite facets.
6.
The diagonals of an " n "-parallelotope intersect at one point and are bisected by this point.
7.
In the case " m " = 1 the parallelotope is reduced to a single vector and its volume its length.
8.
An " n "-vector doesn't necessarily have a shape of a parallelotope this is a convenient visualization.
9.
However, if one wishes to define a function that assigns a volume for any such parallelotope, it should satisfy the following properties:
10.
This formula expresses the fact that the absolute value of the determinant of a matrix equals the volume of the parallelotope spanned by its columns or rows.
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