Extremal polygons with minimal perimeter
Given a lattice in Euclidean space \({\mathbb E}^n\) with Dirichlet-Voronoi cell \(D\), Minkowski's classic result is as follows. A bounded centrally symmetric convex body \(K\) in \({\mathbb E}^n\) centered at \(\mathbf 0\) and with volume \(v(K) > 2^n v(D)\) contains a lattice point different from \(\mathbf 0\). An extremal body with respect to \(L\) is a \(\mathbf 0\)-symmetric closed convex body with volume \(2^n v(D)\) containing no non-zero lattice point in the interior. Minkowski proved that if \(K\) is extremal, then \(K\) is a centrally symmetric polytope and \({\mathbb E}^n\) is tiled by copies of \({1 \over 2} K\) translated by the lattice \(L\). In the paper under review, the following questions are answered in the 2-dimensional case. (1) Which bodies have minimal surface area in the class of extremal bodies for a fixed \(n\)-dimensional lattice? (2) Which bodies have minimal surface area in the class of extremal bodies with volume 1? The answer to the second question is the regular hexagon and the corresponding lattice is the regular triangular lattice. The answer to the first question is a certain hexagon described in the paper.
scientific article; zbMATH DE number 1099690
scientific article