关于正规凸镶嵌的边密度
On the edge densities of normal, convex mosaics
AI总结：
本文研究正规凸镶嵌的最小边密度，证明二维由正六边形取得，高维由正交正三角形闵可夫斯基和取得，三维立方体最优，并找出最小总边长的单位体积多面体。
摘要：
本文研究在$n$维欧几里得空间中，具有单位体积胞腔的凸正规镶嵌族中最小边密度的问题。在论文的第一部分，我们解决了胞腔为$1$维或$2$维镶嵌胞腔的闵可夫斯基和的镶嵌族中的这一问题。我们证明，对于$n=2$，该最小值由具有正六边形胞腔的镶嵌取得，但在任何$n>2$的维度中，这一结论不再成立，此时最小值由胞腔为两两正交的正三角形（可能加上一个线段）的闵可夫斯基和的镶嵌取得。在第二部分，我们研究胞腔为给定凸多面体的平移的$3$维凸镶嵌，并证明在该族中，以立方体为胞腔的镶嵌具有最小边密度。此外，利用我们的方法，在平移可铺满空间的$3$维凸多面体族中，我们找到了具有最小总边长的单位体积多面体。
英文摘要：
In this paper we investigate the problem of finding the minimum edge density in families of convex, normal mosaics with unit volume cells in $n$-dimensional Euclidean space. In the first part of the paper we solve this problem for mosaics whose cells are Minkowski sums of cells of $1$ or $2$-dimensional mosaics. We show that while for $n=2$ this minimum is attained by a mosaic with regular hexagon cells, this is not true in any dimension $n > 2$, where the minimum is attained by a mosaic whose cells are Minkowski sums of pairwise orthogonal regular triangles, and possibly a segment. In the second part we investigate $3$-dimensional convex mosaics whose cells are translates of a given convex polyhedron, and show that within this family, mosaics with cubes as cells have minimum edge density. In addition, using our method, in the family of $3$-dimensional convex polyhedra whose translates tile the space, we find the unit volume polyhedra with minimal total edge length.
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