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Minimal requirements for Minkowski's theorem in the plane II

Published online by Cambridge University Press:  17 April 2009

J.R. Arkinstall
J.R. Arkinstall
Affiliation:
Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia 5001, Australia.
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Abstract

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Let K be a closed convex set in the Euclidean plane, with area A(K), which contains in its interior only one point 0 of the integer lattice. If K has other than one or three chords through 0 of one of the following types, it is shown that A(K) ≤ 4, while if K has three of one type, A(K) ≤ 4.5. The types of chords considered are chords which partition K into two regions of equal area, chords which lie midway between parallel supporting lines of K, and chords such that K is invariant under reflection in them. The results are generalised to any lattice in the plane.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 22 , Issue 2 , October 1980 , pp. 275 - 283
Copyright
Copyright © Australian Mathematical Society 1980

References

[1]Arkinstall, J.R., “Minimal requirements for Minkowski's theorem on the plane I”, Bull. Austral. Math. Soc. 22 (1980), 259–274.CrossRefGoogle Scholar
[2]Scott, P.R., “On Minkowski's theorem”, Math. Mag. 47 (1974), 277.CrossRefGoogle Scholar
[3]Scott, P.R., “Convex bodies and lattice points”, Math. Mag. 48 (1975), 110–112.CrossRefGoogle Scholar