Isoperimetrically Optimal Polygons in the Triangular Grid

Isoperimetrically optimal polygons in the triangular grid with Jordan-type neighbourhood on the boundary

“…This paper is the extended version of [25]. Some of the results of Sections 2 and 4 can also be found there.…”

“…The results of this section can also be found in [25], but for sake of completeness they are fully presented here.…”

“…In this paper we are using these two types of neighbourhood relations. In this paper we recall some results of [1,29] (regarding isoperimetric inequality in the square grid), moreover we give an alternative (and in some sense simpler) proof of the result based on combinatorics ( [25]). Our main results are extensions of these results to the triangular grid with the two types of Jordan neighbourhood.…”

Maximal Fluctuations Around the Wulff Shape for Edge-Isoperimetric Sets in $$\varvec{{\mathbb {Z}}^d}$$: A Sharp Scaling Law

“…In two dimensions, optimal polyominoes and lattice animals are discussed in [10,26]. Indeed, characterizing isoperimetrically optimal polyominoes and polycubes is a classical problem in discrete mathematics and is also considered in [3,23,24,37,42]. We refer to [1,6,9,27] for further results in combinatorics and for optimization problems on graphs.…”

Binary tomography on the isometric tessellation involving pixel shape orientation

“…The natural directions of the tessellation are cached by lanes: their role is very similar to the roles of rows and columns in the rectangular tessellation. The natural shape of the images is hexagon on the triangular grid: every shape has an embedding hexagon [19, 20]. Thus, we consider binary images that have regular hexagonal shape.…”