Isoperimetric problems for tilings, corrigendum

Extract

In [1] I made the following conjecture. If we tile a convex polygon of at most six sides with N convex tiles of areas a₁, …,a_N, then the total perimeter of the tiles is never less than the total perimeter of N regular hexagons of areas a₁, …, a_N. In order to show that the condition of the convexity of the tiles cannot be omitted I constructed a tiling with an equal number of “pentagons” and “heptagons” of perimeters and areas and , and ā₅ and ā₇, respectively. In a letter to me R. Schneider kindly pointed out that, in contrast to the value given in [1], the value of the quotient was equal to 3.7263… This being greater than √8√3 = 3–7224… the tiling given in [1] does not yield the desired counter-example. In what follows we shall construct a tiling for which the respective quotient will turn out to be less than √8√3.