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Acta Sci. Math. (Szeged), 12 (1950), 62–67, see Theorem I and the remarks on page 66.
A convex polygon having at most six sides will be called a hexagon.
My first result, Theorem 2, was obtained in 1947, and was described in seminars in London, Cambridge, Bristol and Princeton in the years 1948–49; its most important consequence was announeed in a paper byJ. H. H. Chalk and myself (J. L. M. S., 23 (1948), 178–187 (179)). Detailed proofs of the results were given in the version of the present paper originally submitted to Acta mathematica.
This inequality is not difficult to prove. By continuity considerations it suffices to prove the inequality in the case whenK is strictly convex. If one considers a lattice Λ with determinantd(K) giving a lattice packing of strictly convex setsK, it follows from the well know theory ofMinkowski (Diophantische Approximationen (Teubner, Berlin 1947), § 4, or seeK. Mahler, Proc. London Math. Soc. (2) 49 (1946), 158) that each setK+x withx in Λ has a boundary point in common with the boundaries of just six of the other sets of this form. If tac-lines are drawn toK through these six points of contact, care being taken to ensure that opposite tac-lines are parallel, they bound an open hexagonH cireumscribingK, and no two of the hexagonsH+x withx in Λ have common points. Thus we see thath(K)≤a(H)≤d(A)=d(K).
Abh. Math. Sem. Hamb. Univ., 10 (1934), 216–230 or seeK. Mahler, Proc. K. Ned. Akad. v. Wet. (Amsterdam), 50 (1947), 692–703.
An open setK is said to be strictly convex if it is such that, for every pair of distinct pointsa andb on the boundary ofK, every inner pointc of the line segment joininga tob is inK.
Compare withB. Segre andK. Mahler, Amer. Math. Monthly, 51 (1944), 261–270, § 5.
W. Blaschke,Kreis und Kugel (Leipzig, 1916), § 18.
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An erratum to this article is available at http://dx.doi.org/10.1007/BF02546391.
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Rogers, C.A. The closest packing of convex two-dimensional domains. Acta Math. 86, 309–321 (1951). https://doi.org/10.1007/BF02392671
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DOI: https://doi.org/10.1007/BF02392671