Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-25T08:23:38.404Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Irreducibility of Convex Bodies

Published online by Cambridge University Press:  20 November 2018

A. C. Woods
A. C. Woods*
Affiliation:
Tulane University of Louisiana
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We select a Cartesian co-ordinate system in ndimensional Euclidean space Rn with origin 0 and employ the usual pointvector notation.

By a lattice Λ in Rn we mean the set of all rational integral combinations of n linearly independent points X1, X2, … , Xn of Rn. The points X1 X2, … , Xn are said to form a basis of Λ. Let {X1, X2, … , Xn) denote the determinant formed when the co-ordinates of Xi are taken in order as the ith row of the determinant for i = 1,2, … , n. The absolute value of this determinant is called the determinant d(Λ) of Λ. It is well known that d(Λ) is independent of the particular basis one takes for Λ.

A star body in Rn is a closed set of points K such that if X ∈ K then every point of the form tX where — 1 < t < 1 is an inner point of K.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 256 - 261
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Mahler, K., Lattice points in n-dimensional star bodies II, Reducibility theorems, Proc. Nederl. Akad. Wetensch., 49 (1946), 331–43.Google Scholar
2. Mahler, K. On irreducible convex domains, Proc. Nederl. Akad. Wetensch., 50 (1947), 98–107.Google Scholar
3. Mahler, K. Lattice points in n-dimensional star bodies, I, Existence theorems, Proc. Roy. Soc. London, Ser. A, 187 (1946), 151-87.Google Scholar
4. Ollerenshaw, K., Irreducible convex bodies, Quart. J. Math., Oxford (2), 4 (1953), 293–302.Google Scholar
5. Rogers, C. A., A note on irreducible star bodies, Proc. Nederl. Akad. Wetensch., 50 (1947), 868–72.Google Scholar
6. Whitworth, J. V., On the densest packing of sections of a cube, Ann. Mat. Pura Appl., Ser. 4, 27 (1948), 29–37.Google Scholar