Question

Show that the conditions of Minkowski's Lattice-Point Theorem cannot be relaxed by sketching

(a) a centrally symmetric convex subset of R² of area 4 ,
(b) a convex subset of R² of infinite area,
(c) a centrally symmetric subset of R² of infinite area, each of which contains no non-zero point with integer co-ordinates.

Answer 1

To show that the conditions of Minkowski's Lattice-Point Theorem cannot be relaxed, we can sketch examples of convex subsets that do not contain any non-zero points with integer coordinates.

Step-by-step explanation:

In order to show that the conditions of Minkowski's Lattice-Point Theorem cannot be relaxed, we need to sketch examples of convex subsets of R² that do not contain any non-zero points with integer coordinates.

For part (a), we can sketch a centrally symmetric convex subset of R² with an area of 4, such as a square with side length 2. The coordinates of the vertices of the square would be (-1,-1), (-1,1), (1,1), and (1,-1), and none of these points have integer coordinates.

For part (b), we can sketch a convex subset of R² with infinite area, such as a line segment with slope greater than 1. No points on this line segment would have integer coordinates.

Similarly, for part (c), we can sketch a centrally symmetric subset of R² with infinite area, such as a parabola opening upwards. Again, no points on this parabola would have integer coordinates.