论文标题
在几乎正交晶格的几何形状上
On the geometry of nearly orthogonal lattices
论文作者
论文摘要
在[4]中正式定义了几乎正交的晶格，还讨论了它们在图像压缩中的应用。 The idea of ``near orthogonality" in $2$-dimensions goes back to the work of Gauss. In this paper, we focus on well-rounded nearly orthogonal lattices in~$\mathbb R^n$ and investigate their geometric and optimization properties. Specifically, we prove that the sphere packing density function on the space of well-rounded lattices in dimension $n\geq 3$ does not have any local在几乎正交的集合上，在那里只有一个本地最低限度：在整个晶格〜$ \ Mathbb z^n $中，我们还表明，尽管它包含〜$ n \ geq 3 $的几乎正交套件，但它包含多个eutactic（甚至是强烈的eutactic）lattementes lattementes lattementes latters latt intimention。包装密度功能不一定是局部最大值或最小值，即使在全面的晶格中，我们还证明了（弱）几乎是正交的晶格，〜$ \ $ \ Mathbb r^n $不超过〜$ 4N-2 $的最小矢量（可能有任何较小的数字），并建立了这些Latters。
Nearly orthogonal lattices were formally defined in [4], where their applications to image compression were also discussed. The idea of ``near orthogonality" in $2$-dimensions goes back to the work of Gauss. In this paper, we focus on well-rounded nearly orthogonal lattices in~$\mathbb R^n$ and investigate their geometric and optimization properties. Specifically, we prove that the sphere packing density function on the space of well-rounded lattices in dimension $n\geq 3$ does not have any local maxima on the nearly orthogonal set and has only one local minimum there: at the integer lattice~$\mathbb Z^n$. Further, we show that the nearly orthogonal set cannot contain any perfect lattices for~$n \geq 3$, although it contains multiple eutactic (and even strongly eutactic) lattices in every dimension. This implies that eutactic lattices, while always critical points of the packing density function, are not necessarily local maxima or minima even among the well-rounded lattices. We also prove that a (weakly) nearly orthogonal lattice in~$\mathbb R^n$ contains no more than~$4n-2$ minimal vectors (with any smaller even number possible) and establish some bounds on coherence of these lattices.
