Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-17T17:02:17.732Z Has data issue: false hasContentIssue false
  • English
  • Français

On the maximal circumradius of a planar convex set containing one lattice point

Published online by Cambridge University Press:  17 April 2009

Poh W. Awyong  and
Paul R. Scott
Poh W. Awyong
Affiliation:
Department of Pure MathematicsThe University of AdelaideSouth Australia 5005 e-mail: [email protected]@maths.adelaide.edu.au
Paul R. Scott
Affiliation:
Department of Pure MathematicsThe University of AdelaideSouth Australia 5005 e-mail: [email protected]@maths.adelaide.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We obtain a result about the maximal circumradius of a planar compact convex set having circumcentre O and containing no non-zero lattice points in its interior. In addition, we show that under certain conditions, the set with maximal circumradius is a triangle with an edge containing two lattice points.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 52 , Issue 1 , August 1995 , pp. 137 - 151
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Eggleston, H.G., Convexity (Cambridge University Press, 1958).CrossRefGoogle Scholar
[2]Eggleston, H.G., ‘Notes on Minkowski geometry (I): Relations between the circumradius, diameter, inradius and minimal width of a convex set’, J. London Math. Soc. 33 (1958), 76–81.Google Scholar
[3]Maxwell, E.A., Geometry for advanced pupils (Oxford University Press, 1949).Google Scholar
[4]Henk, M. and Tsintsifas, G.A., ‘Some inequalities for planar convex figures’, Elem. Math. 49 (1994), 120–124.Google Scholar
[5]Scott, P.R., ‘Two inequalities for convex sets in the plane’, Bull. Austral. Math. Soc. 19 (1978), 131–133.CrossRefGoogle Scholar
[6]Scott, P.R., ‘A family of inequalities for convex sets’, Bull. Austral Math. Soc. 20 (1979), 237–245.CrossRefGoogle Scholar
[7]Scott, P.R., ‘Further inequalities for convex sets with lattice point constraints in the the plane’, Bull. Austral. Math. Soc. 21 (1980), 7–12.CrossRefGoogle Scholar
[8]Scott, P.R., ‘Sets of constant width and inequalities’, Quart. J. Math. Oxford Ser. (2) 32 (1981), 345–348.CrossRefGoogle Scholar
[9]Yaglom, I.M. and Boltyanskii, V.G., Convex figures, Translated by Kelly, P.J. and Walton, L.F. (Holt, Rinehart and Winston, New York, 1961).Google Scholar