Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-28T14:58:05.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Circumradius-diameter and width-inradius relations for lattice constrained convex sets

Published online by Cambridge University Press:  17 April 2009

Poh Wah Awyong  and
Paul R. Scott
Poh Wah Awyong
Affiliation:
Division of Mathematics, National Institute of Education, 469 Bukit Timah Road, Singapore 259756 e-mail: [email protected]
Paul R. Scott
Affiliation:
Department of Pure Mathematics, The University of Adelaide, South Australia 5005 e-mail: [email protected]
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.

Let K be a planar, compact, convex set with circumradius R, diameter d, width w and inradius r, and containing no points of the integer lattice. We generalise inequalities concerning the ‘dual’ quantities (2R − d) and (w − 2r) to rectangular lattices. We then use these results to obtain corresponding inequalities for a planar convex set with two interior lattice points. Finally, we conjecture corresponding results for sets containing one interior lattice point.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 59 , Issue 1 , February 1999 , pp. 147 - 152
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Awyong, P.W., ‘An inequality relating the circumradius and diameter of two-dimensional lattice-point-free convex bodies’, Amer. Math. Monthly (to appear).Google Scholar
[2]Scott, P.R., ‘Two inequalities for convex sets in the plane’, Bull. Austral. Math. Soc. 19 (1978), 131–133.CrossRefGoogle Scholar
[3]Scott, P.R., ‘A family of inequalities for convex sets’, Bull. Austral. Math. Soc. 20 (1979), 237–245.Google Scholar
[4]Scott, P.R., ‘Sets of constant width and inequalities’, Quart. J. Math. Oxford Ser. (2) 32 (1981), 345–348.Google Scholar
[5]Scott, P.R., ‘On planar convex sets containing one lattice point’, Quart. J. Math. Oxford Ser. (2) 36 (1985), 105–111.Google Scholar
[6]Vassallo, S., ‘A covering problem for plane lattices’, Geom. Dedicata 43 (1992), 321–335.Google Scholar