Correction to my paper “Plane fundamental domains with minimal perimeters”

Abstract

In this short notice I give the correction of the table for Theorem 1 in [1]. The aim of this short notice is to correct some errors in the paper “Plane fundamental domains with minimal perimeters” which was published in this journal [1]. After thorough study it was Endre Makai Jr. who discovered some problems in the table belonging to Theorem 1. He realized that at some critical cases where the optimal solution depends on the affine parameters the given perimeters do not coincide albeit the perimeter-function should be continuous. He immediately asked me about the reasons and I gave him the following correction. For completeness I repeat the theorem, too. Theorem 1. There are 13 plane crystallographic groups, where the fundamental domain, having minimal perimeter, is unique. For the groups pg, cm, pmg and pgg there are two possible planigon types which serve the minimal perimeter according to the affine parameters. Our following table contains the complete list of the plane groups with the optimal planigon types, minimal perimeters and, if necessary, with conditions for the affine parameters. The problems were caused partly by miscount (the case of pg), partly by miswrites (cases cm, pmg). I am very grateful to Endre Makai Jr. for his valuable observation. Mathematics subject classification number: 52B60, 52C20.