Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 41, No. 2, pp. 329-334 (2000)

On Parametric Density of Finite Circle Packings

Achill Schürmann

Fachbereich Mathematik, Universität Siegen, Postfach 101240, D-57068 Siegen 21, Germany

Abstract: The parametric density of a finite circle packing $X_n+D$ is defined as ratio of areas ${n \cdot A(D)}/{A\left( conv(X_n+\rho D)\right)}$. We show that densest finite lattice packings are attained in a critical lattice. Further, we give an upper bound for the density with parameters $\rho\leq 3.232\dots$ which is attained by Groemerpackings. Moreover, we show that the given bound holds for arbitrary finite circle packings and parameters less than $1$ as a consequence of known results by H. Groemer [A] and G. Wegner [B]. \item{[A]} Groemer, H.: Über die Einlagerung von Kreisen in einen konvexen Bereich. Math. Z. 73 (1960), 285-294. \item{[B]} Wegner, G.: Über endliche Kreispackungen in der Ebene. Studia Sci. Math. Hung. 21 (1986), 1-28.

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