Paul R. Scott

On convex lattice polygons

A lattice is a (rectangular) grid of points, usually pictured as occurring at the intersections of two orthogonal sets of parallel, equally spaced lines. The concept of Lattice theory and the area measurement is explained.

The Fascination of the Elementary

We begin with the humble geoboard, a teaching aid used (at least) in Australian schools. It is a square board with nails (or golf tees) placed in a square grid formation. Various polygons can be constructed by wrapping a rubber band around the nails (see Figure 1). This illustrates two basic concepts: The integer lattice in the plane (or in general dimension) is the set of points having integral coordinates. (More general lattices can be obtained by applying a linear transformation to the integer lattice; many of our results adapt easily to these lattices.) A lattice polygon is a polygon which has a lattice point at every vertex.

A family of inequalities for convex sets

Let K be a bounded, closed convex set in the euclidean plane. We denote the diameter, width, perimeter, area, inradius, and circumradius of K by d, w, p, A, r , and R respectively. We establish a number of best possible upper bounds for (w-2r)d , (w-2r)R , (u-2r)p , (w-2r)A in terms of w and r . Examples are:

Properties of axial diameters

Λ is a lattice and K a bounded, open, convex set in E n . An axial diameter of K is the maximal length X i , of chords of K parallel to the i th lattice basis vector (1 ≤ i ≤ n ). A number of properties of the axial diameters are developed. For sets K containing just one lattice point, an inequality is established; when Λ is the integral lattice, this inequality takes the form .

Modifying Minkowski's theorem

Abstract At the turn of the century, Minkowski published his famous “convex body” theorem which became the basis for the geometry of numbers. Suppose that Λ is a lattice in Euclidean n-space, En, having determinant d(Λ). Now Minkowskis theorem states that if K is a convex body which is symmetric about the origin O, and if K contains no nonzero points of the lattice Λ, then the volume V(K) of K satisfies V(K) ≤ 2n d(Λ). In spite of its simple nature, Minkowskis theorem is a powerful and important result. Since he theorem first appeared, there have been a surprisingly large number of modifications and variations published. A number of these will be discussed, particularly some of the more recent discoveries and unproved conjectures.

New inequalities for planar convex sets with lattice point constraints

We obtain new inequalities relating the inradius of a planar convex set with interior containing no point of the integral lattice, with the area, perimeter and diameter of the set. By considering a special sublattice of the integral lattice, we also obtain an inequality concerning the inradius and area of a planar convex set with interior containing exactly one point of the integral lattice.

An isodiametric problem with lattice-point constraints

In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space ({Bbb R}^d) containing no interior non-zero point of a lattice L is studied. It is shown that the intersection of a suitable ball with the Dirichlet-Voronoi cell of 2L is extremal, i.e., it has minimum diameter among all bodies with the same volume. It is conjectured that these sets are the only extremal bodies, which is proved for all three dimensional and several prominent lattices.

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

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 (2 R − d ) and ( w − 2 r ) 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.

Width-diameter relations for planar convex sets with lattice point constraints

We obtain an inequality concerning the width and diameter of a planar convex set with interior containing no point of the rectangular lattice. We then use the result to obtain a corresponding inequality for a planar convex set with interior containing exactly two points of the integral lattice.

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

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.