Minkowski's Convex Body Theorem
Suppose
Before diving deeper, let's first clearly understand some terms used in the theorem..
Convex
A set
An example for a set in
Symmetric about the origin
A set
Lattice
A lattice is an array of points that differ in equal intervals in any dimension. The most widely used lattice is the well-known "integer lattice ( )". For example, non-integer lattices, are hexagonal and parallelogram lattices in the Euclidean Plane.
Determinant of a lattice
The determinant of a lattice L is the volume of the smallest area enclosed by the lattice point. This is represented by . For example, the determinant of the lattices in the above will just be the area of "1 hexagon" and "1 parallelogram formed by 4 adjacent lattice points" respectively.
Now, we understand what the theorem says completely. Let's move on to building the foundation for our proof.
Blichfeldt's Theorem
If R is a bounded set in with area greater than 1, then R contains two distinct points and such that the point is an integer point in
Proof: Let . For every point , define be the translation of along the line segment with endpoints and . Note that will be the only integer point in Since, is bounded, we know that there will be a finite number of points such that is non empty. Let be the translation of back to along the line segment with endpoints and . Since the translations are plane isometry, the area of the sets during translation will be preserved, thus, . We have,
This means that that all the subsets are stacked one upon another after the translation. Since, the sum of area is greater than 1, we can say that, after the translation, there will be at least integer points and such that the areas of and overlap. That is, . Define . Let and by definition. We have,
Thus, there are two distinct points in R and , whose difference in co-ordinates is an integer point.
Remark: Note that this theorem is also true for "non-integer lattices", say . The only difference would be that should have an area greater than . The proof will be similar to this.
This is an extremely helpful theorem which almost finishes the proof of "Minkowski's Theorem". Now, let's move on to prove a simpler version of "Minkowski's Convex Body Theorem".
Simpler version
Let be a convex region in that is symmetric about the origin and has an area greater than . Then contains an integer point other than the origin.
Proof: Define the set . Since is just scaled down by a factor of , it is also convex and symmetric about the origin. Moreover, . By Blichfeldt's Theorem, there exists two distinct points such that is an integer point. Note that . Since, is symmetric about the origin, . We know that is convex. Thus, every point on the line segment with endpoints and lies inside . Therefore, their midpoint also lies in . We have
Note that the symmetric condition yield that along with , the point must also lie in . Thus, there are at least 3 integer points in .
Minkowski's Theorem for Arbitrary Lattices
For, arbitrary lattices, Minkowski's Theorem can be proved with the help of "Pick's Theorem". Although this will not be discussed in this blog, you can refer to its wonderful proof here
Applications of Minkowski's Theorem
The two squares theorem: For a prime , we can always find some , such that
Proof: By, Fermat's Christmas Theorem, is a quadratic residue modulo . Thus, there exists some such that . Let and . Define a lattice such that , , and are all lattice points. Note that these four points form a parallelogram with area . Thus, . \newline
Now consider a set in . We have,
By "Minkwoski's Theorem", we know that contains some point . Since, can be written as a linear combination of and , let , which implies
Since, , we must have . Hence, proved.
Post by Akshat Pandey.
About the guest blogger: Hi, I am Akshat Pandey, an 11th grader. I am an olympiad math freak who also enjoys playing the piano, badminton and table tennis.
Post modified and set up by Sunaina Pati. This post is part of Akshat's lecture notes for Sophie fellowship's WeMP.
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