AI 中文总结
研究平面到两个不同单元的周期性划分以最小化各向异性\(\ell_1\)周长，对矩形格计算等周轮廓并分类极小值，在所有平面格上最小化得出毕达哥拉斯双铺砌实现\(\ell_1\)等周轮廓，还证明了相关极限划分的局部\(\ell_1\)等周性。
AI 中文摘要
我们研究平面到两个不同单元的周期性划分，以最小化各向异性\(\ell_1\)周长。对于矩形格\(G\)，我们明确计算了\((G,\ell_1)\)等周轮廓并对所有极小值进行分类。当一个单元面积小时，最优铺砌由一个正方形和一个缺角矩形生成，其余情况下由两个相邻矩形生成。在所有可能的平面格上进一步最小化，我们表明\(\ell_1\)等周轮廓由共享一个顶点的两个轴对齐正方形的毕达哥拉斯双铺砌实现。此配置是唯一的，除非两个单元面积相同。最后，我们证明通过将一个体积发送到无穷大得到的极限（非周期性）划分是局部\(\ell_1\)等周的。
英文摘要
We study periodic partitions of the plane into two distinct cells minimizing the anisotropic $\ell_1$-perimeter. For a rectangular lattice $G$, we compute explicitly the $(G,\ell_1)$-isoperimetric profile and classify all minimizers. When one cell has small area, the optimal tiling is generated by a square and a chipped rectangle, while in the remaining regime the tiling is generated by two adjacent rectangles. Further minimizing over all possible planar lattices, we show that the $\ell_1$-isoperimetric profile is attained by the Pythagorean double tiling of two axis-aligned squares sharing a vertex. This configuration is unique unless the two cells are assigned the same area. Finally, we prove that the limiting (non periodic) partitions obtained by sending one volume to infinity are locally $\ell_1$-isoperimetric.
Comments20 pages, 8 figures
