Minkowski Geometry of Numbers

I bought a two volume book on Minkowski's Geometry of Numbers. It's quite old and a bit hard to read but I've studied the first couple of pages where it introduces the idea of a convex body and then proves a theorem about it.

A convex body is defined by a real valued function $\phi(x,y,z)$, the surface of the standard body is the set of points $\phi(x,y,z) = 1$ and the volume of the standard body is $\phi(x,y,z) \le 1$. The function is such that any "homothetic" dilation of the body by a factor of $t$ is given by $\phi(x,y,z) = t$ - this would be called the $t$-body. To start with lets have the origin, dilation and symmetric properties:

$\phi(0,0,0) = 0$ $\phi(tx,ty,tz) = t\phi(x,y,z)$ $\phi(-x,-y,-z) = \phi(x,y,z)$

A simple example of a body might be the sphere: $\phi(x,y,z) = x^2 + y^2 + z^2$

Convexivity and the triangle inequality

Let $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ be two convex bodies with radial distances $t_1$ and $t_2$ so that $(x_1/t_1, y_1/t_1, z_1/t_1)$ and $(x_2/t_2, y_2/t_2, z_2/t_2)$ both lie on the surface of the standard body. By the formula for weighted center of mass $\frac{\sum x_i m_i}{\sum m_i}$ the point $(\frac{x_1+x_2}{t_1+t_2}, \frac{y_1+y_2}{t_1+t_2}, \frac{y_1+y_2}{t_1+t_2})$ lies in the standard body, i.e. $\phi(\frac{x_1+x_2}{t_1+t_2}, \frac{y_1+y_2}{t_1+t_2}, \frac{y_1+y_2}{t_1+t_2}) \ge 1$ now dilating $\phi(x_1+x_2, y_1+y_2, y_1+y_2) \ge t_1 + t_2$ hence we have the triangle inequality for a convex body: $\phi(x_1+x_2, y_1+y_2, y_1+y_2) \ge \phi(x_1, y_1, z_1) + \phi(x_2, y_2, z_2)$

Transformations

If we make the substitution

• $x = \lambda \xi + \lambda' \eta + \lambda'' \zeta$
• $y = \mu \xi + \mu' \eta + \mu'' \zeta$
• $z = \nu \xi + \nu' \eta + \nu'' \zeta$

(whose determinant must be nonzero) then $\phi(x,y,z)$ becomes $F(\xi,\eta,\zeta)$ which is again a convex body.

Lattices, accordant bodies

Minkowski's theory is all about the lattice points that must lie inside these convex bodies. An important concept is the accordant-body: Let us take some lattic in the background and a body $\phi$ over it. Dilate it very very small so that it contains no lattice points other than the origin, then consider scaling it up until it just touches the first lattice point. Call this dilation $M$ and we have the accordant $M$-body.

If we now consider laying a collection of $M/2$-bodies with their centers on each lattice point then they would all contain only their own center (a lattice point) and just touch the boundary of some of their neighbouring bodies.

This concept is useful in proving the first theorem: A convex symmetric body with volume $8$, having its center on a lattice point will always contain another lattice point. In this case the lattice must have determinant 1.

To see this take a body with volume $J$ and dilate it to an $M/2$-body, this will have volume $(\frac{M}{2})^3J$. Now tile space with the parrellelpiped volumes of the lattice, each of these units has volume 1. The bodies at each lattice point are covered by these and may have some gaps so we have $(\frac{M}{2})^3J \le 1$ thus $M^3J \le 8$

The inequality states that the volume of the $M-body$ is smaller than $8$. This proves that any dilation of this body with volume larger than 8 contains a second lattice point other than its origin. That is true because these bodies are concentric.

End

I would like to keep studying this theory and see what number theory results can be found with it.