关于标准格（英文）

On the Standard Lattices

欧氏空间中的一个格如果有一组基所包含的向量的范数对应等于它的短线段列长度,则称这个格是标准的.本文证明了在L^2范数下,所有n维的格都是标准的当且仅当n不超过4.同时,我们证明了在任意范数下每个1维和2维的格都是标准格.我们也给出了在L^1范数下不低于3维的非标准格的例子.

A lattice in the Euclidean space is standard if it has a basis consisting vectors whose norms equal to the length in its successive minima. In this paper, it is shown that with the L^2 norm all lattices of dimension n are standard if and only if n 4. It is also proved that with an arbitrary norm, every lattice of dimensions 1 and 2 is standard. An example of non-standard lattice of dimension n 3 is given when the lattice is with the L^1 norm.