Note on an Inequality of Wegner

G. Wegner [12] gave a geometric characterization of all so–called Groemer packing of n ≥ 2 unit discs in E2 that are densest packings of n unit discs with respect to the convex hull of the discs. In this paper we provide a number theoretic characterization of all n satisfying that such a “Wegner packing” of n unit discs exists, and show that the proportion of these n is 23 24 among all natural numbers. Acknowledgement: We are grateful to J.C. Lagarias for helpful discussions, and to an unknown referee whose remarks considerably improved the paper. ∗Supported by OTKA grants T 042769, 043520 and 049301, and by the Marie Curie TOK project DiscConvGeo. †Supported by OTKA grants T 025617, 038396 and 042750.