Extremal polygons with minimal perimeter
type: Item
instance of: scholarly article
title: Extremal polygons with minimal perimeter
zbMATH Open document ID: 0880.52012
DOI: 10.1023/A:1004276624604
published in: Periodica Mathematica Hungarica
publication date: 1998-01-05T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 52C17
zbMATH DE Number: 1099690
zbMATH Keywords: close packing of balls
zbMATH Keywords: thick point sets
zbMATH Keywords: spherical space
author: Ákos G. Horváth
reviewed by: Andrew Vince
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1099690