2023/12/13 by Kadlicskó, Máté, Lángi, Zsolt, Lyu, Shanxiang
#52A38 #52A40 #52C22 #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2312.08050
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.