Inequalities for Lattice Constrained Planar Convex Sets

Volume 3, Issue 2, 2002

Article 23

INEQUALITIES FOR LATTICE CONSTRAINED PLANAR CONVEX SETS

4/38 BEAUFORT STREET, ALDERLEY, QUEENSLAND 4051, AUSTRALIA.

DEPARTMENT OF PURE MATHEMATICS,
UNIVERSITY OF ADELAIDE, S.A. 5005
AUSTRALIA.
E-Mail: pscott@maths.adelaide.edu.au

Received 2000 ; 28 November, 2001.
Communicated by: C.E.M. Pearce

ABSTRACT. Every convex set in the plane gives rise to geometric functionals such as the area, perimeter, diameter, width, inradius and circumradius. In this paper, we prove new inequalities involving these geometric functionals for planar convex sets containing zero or one interior lattice point. We also conjecture two results concerning sets containing one interior lattice point. Finally, we summarize known inequalities for sets containing zero or one interior lattice point.

Key words:
Planar Convex Set, Lattice, Lattice Point Enumerator, Lattice-Point-Free, Sublattice, Area, Perimeter, Diameter, Width, Inradius, Circumradius.

2000 Mathematics Subject Classification:
52A10, 52A40, 52C05, 11H06.

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