Minkowski's second theorem explained

In mathematics, Minkowski's second theorem is a result in the geometry of numbers about the values taken by a norm on a lattice and the volume of its fundamental cell.

Setting

Let be a closed convex centrally symmetric body of positive finite volume in -dimensional Euclidean space . The gauge^[1] or distance^{[2] [3]} Minkowski functional attached to is defined by$$g(x)\; =\; \backslash inf\; \backslash left\backslash \; .$$

Conversely, given a norm on we define to be$$K\; =\; \backslash left\backslash \; .$$

Let be a lattice in . The successive minima of or on are defined by setting the -th successive minimum to be the infimum of the numbers such that contains linearly-independent vectors of . We have .

Statement

The successive minima satisfy^{[4] [5] [6]} $$\backslash frac\; \backslash operatorname\backslash left(\backslash mathbb^n/\backslash Gamma\backslash right)\; \backslash le\; \backslash lambda\_1\backslash lambda\_2\backslash cdots\backslash lambda\_n\; \backslash operatorname(K)\backslash le\; 2^n\; \backslash operatorname\backslash left(\backslash mathbb^n/\backslash Gamma\backslash right).$$

Proof

A basis of linearly independent lattice vectors can be defined by .

The lower bound is proved by considering the convex polytope with vertices at, which has an interior enclosed by and a volume which is times an integer multiple of a primitive cell of the lattice (as seen by scaling the polytope by along each basis vector to obtain -simplices with lattice point vectors).

To prove the upper bound, consider functions sending points in $K$ to the centroid of the subset of points in $K$ that can be written as $x\; +\; \backslash sum\_^\; a\_i\; b\_i$ for some real numbers $a\_i$. Then the coordinate transform $$x\text{'}\; =\; h(x)\; =\; \backslash sum\_^\; (\backslash lambda\_i\; -\backslash lambda\_)\; f\_i(x)/2$$ has a Jacobian determinant $J\; =\; \backslash lambda\_1\; \backslash lambda\_2\; \backslash ldots\; \backslash lambda\_n/2^n$. If $p$ and $q$ are in the interior of $K$ and $p-q\; =\; \backslash sum\_^k\; a\_i\; b\_i$(with $a\_k\; \backslash neq\; 0$) then $$(h(p)\; -\; h(q))\; =\; \backslash sum\_^k\; c\_i\; b\_i\; \backslash in\; \backslash lambda\_k\; K$$ with $c\_k\; =\; \backslash lambda\_k\; a\_k\; /2$, where the inclusion in $\backslash lambda\_k\; K$ (specifically the interior of $\backslash lambda\_k\; K$) is due to convexity and symmetry. But lattice points in the interior of $\backslash lambda\_k\; K$ are, by definition of $\backslash lambda\_k$, always expressible as a linear combination of $b\_1,\; b\_2,\; \backslash ldots\; b\_$, so any two distinct points of $K\text{'}\; =\; h(K)\; =\; \backslash$ cannot be separated by a lattice vector. Therefore, $K\text{'}$ must be enclosed in a primitive cell of the lattice (which has volume $\backslash operatorname(\backslash R^n/\backslash Gamma)$), and consequently $\backslash operatorname\; (K)/J\; =\; \backslash operatorname(K\text{'})\; \backslash le\; \backslash operatorname(\backslash R^n/\backslash Gamma)$.

References

• Book: Cassels, J. W. S. . J. W. S. Cassels . An introduction to Diophantine approximation . Cambridge Tracts in Mathematics and Mathematical Physics . 45 . . 1957 . 0077.04801 .
• Book: Cassels, J. W. S. . J. W. S. Cassels . An Introduction to the Geometry of Numbers . Classics in Mathematics . . Reprint of 1971 . 1997 . 978-3-540-61788-4 .
• Book: Nathanson, Melvyn B. . Additive Number Theory: Inverse Problems and the Geometry of Sumsets . 165 . . . 1996 . 0-387-94655-1 . 0859.11003 . 180–185 .
• Book: Schmidt, Wolfgang M. . Wolfgang M. Schmidt . Diophantine approximations and Diophantine equations . Lecture Notes in Mathematics . 1467 . . 1996 . 2nd . 3-540-54058-X . 0754.11020 . 6 .
• Book: Siegel, Carl Ludwig . Carl Ludwig Siegel . Lectures on the Geometry of Numbers . . 1989 . 3-540-50629-2 . Komaravolu S. Chandrasekharan . Komaravolu S. Chandrasekharan . 0691.10021 .

Notes and References

1. Siegel (1989) p.6
2. Cassels (1957) p.154
3. Cassels (1971) p.103
4. Cassels (1957) p.156
5. Cassels (1971) p.203
6. Siegel (1989) p.57