Parallelotope

A set of points whose radius vectors have the form

$$ \mathbf h = \sum _ { i=1 } ^ { p } x ^ {i} \mathbf a _ {i} , $$

with the possible values $ 0 \leq x ^ {i} \leq 1 $, $ 1 \leq i \leq p $. Here $ \mathbf a _ {1} \dots \mathbf a _ {p} $ are fixed vectors in an affine space $ A $ of dimension $ n $. They are called the generators of the parallelotope and coincide with some of the edges of the parallelotope. All remaining edges of the parallelotope are parallel to them. If the generators of the parallelotope are linearly independent (dependent), then the parallelotope is called $ p $-dimensional, or non-degenerate (degenerate). Degenerate parallelotopes are parallel projections of some $ p $-dimensional parallelotope onto a plane of dimension $ k \leq p- 1 $. A non-degenerate parallelotope determines a supporting $ p $-dimensional plane. Such a parallelotope for $ p= 2 $ is a parallelogram, and for $ p= 3 $ is a parallelopipedon.

Two non-degenerate parallelotopes are said to be parallel if their supporting planes are parallel. For parallel parallelotopes it is possible to compare their $ p $-dimensional "volume" (although there need not be a metric in $ A $). For the numerical measure of the ratio of the $ p $-dimensional "volume" of the parallelotope with generators $ \mathbf a _ {1} \dots \mathbf a _ {p} $ to the $ p $-dimensional "volume" of the parallel parallelotope with generators $ \mathbf b _ {1} \dots \mathbf b _ {p} $, the scalar $ \mathop{\rm det} ( x _ {j} ^ {i} ) $ is used, where $ ( x _ {j} ^ {i} ) $ is the $ ( p \times p) $-matrix which transforms $ ( \mathbf b _ {1} \dots \mathbf b _ {p} ) $ to $ ( \mathbf a _ {1} \dots \mathbf a _ {p} ) $, i.e.

$$ \mathbf a _ {j} = \sum _ { i=1 } ^ { p } x _ {j} ^ {i} \mathbf b _ {i} ,\ \ 1 \leq j \leq p. $$

If an inner product is defined in $ A $, then the square of the $ p $- dimensional volume of the parallelotope with generators $ \mathbf a _ {1} \dots \mathbf a _ {p} $ is equal to the determinant of the $ ( p \times p) $-dimensional Gram matrix with entries $ ( \mathbf a _ {i} , \mathbf a _ {j} ) $ (cf. also Gram determinant).

The concept of a parallelotope is closely connected with the concept of a poly-vector.

References

[1]	P.A. Shirokov, "Tensor calculus. Tensor algebra" , Kazan' (1961) (In Russian)
[2]	D.V. Beklemishev, "A course of analytical geometry and linear algebra" , Moscow (1971) (In Russian)
[3]	C. Pisot, M. Zamansky, "Mathématiques générales: algèbre-analyse" , Dunod (1966)

Comments

Parallelotopes are special types of zonotopes (cf. Zonohedron). They play a basic role in the geometry of numbers and in the theory of lattice covering and packing.

References

[a1]	P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint)
[a2]	B. Grünbaum, "Convex polytopes" , Wiley (1967)