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On lattice points in a convex decagon

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Acta Mathematica

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  1. We use vector notation: thusu 1 X 1+u 2 X 2=(u 1 x 1+u 2 x 2,u 1 y 1+u 2 y 2), and in particular −X 1=(−x 1, −y 1). The determinant ofX 1 andX 2 is denoted by (X 1,X 2)=x 1 y 2−x 2 y 1.

  2. K. Mahler, The Theorem of Minkowski-Hlawka, Duke Mathematical Journal, 14 (1946), 611–621, Lemma 2.

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  3. K. Reinhardt, Über die dichteste gitterförmige Lagerung congruenter Bereiche, und eine besondere Art convexer Curven, Abh. aus dem Math. Seminar der Hamburgische Univ 9 (1933), 216–230. With regard to the smoothed octagon, Reinhardt said: «Die Frage nach den Bereichen dünnster dichtester Lagerung läuft offenbar darauf hinaus, diejenige Kurve (oder diejenigen Kurven), der von uns betracteten Art zu finden, welche bei gegebenem einbeschriebenem etwa regulärem Sechseck eine möglichst kleine Fläche umschliesst. — Bei unseren Bereichen kommt diejenige Figur in Betracht, welche aus einem regelmässigen Achteck entsteht, wenn man jede Ecke durch diejenige Hyperbel abschneidet, die die beiden anstossenden Seiten berührt, und die beiden wieder an diese grenzenden Seiten zu Asymptoten hat.» We call this figure the smoothed octagon.

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  4. K. Mahler, On the minimum determinant and the circumscribed hexagons of a convex domain. Proc. Academy Amsterdam 50 (1947), 692–703, p. 694. This paper will henceforth be referred to asM.

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  5. M, p. 698; p. 702.

  6. Here, as elsewhere, the same letter is used to denote a plane domain and its area.

  7. In order to save space, only four places of decimals are given, but the calculations were actually carried out with greater accuracy.

  8. The bracket denotes again the determinant ofA andB.

  9. See footnote on p. 347.

  10. We are greatly indebted to Mr. D. F. Ferguson, M. A. for helping us with most of the numerical work of this paper.

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  1. Manchester

    Walter Ledermann & Kurt Mahler

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  1. Walter Ledermann
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  2. Kurt Mahler
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Ledermann, W., Mahler, K. On lattice points in a convex decagon. Acta Math. 81, 319–351 (1949). https://doi.org/10.1007/BF02395026

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