Modular Arithmetic/Minkowski's Convex Body Theorem

A convex set is a set of points with the property that given any two points in the set, the straight line joining them lies entirely within the set. Intuitively, this means that the set is connected (so that you can pass between any two points without leaving the set) and has no dents in its perimeter.

Examples of a convex set are a circle, a square and a triangle (assuming that the sets consist of the interiors as well as the circumference or perimeter).

A set is symmetric about a point X if, given any point Y in the set, the point Z lying on the line through X and Y, on the opposite side of X to Y and with ZX = XY, is also in the set.

The Theorem

Blichfeldt's Lemma

Just to be clear, Blichfeldt's lemma isn't so much an application of the pigeonhole principle as an extension of it. Like the pigeonhole principle it is saying that "if you have too much of something it won't all fit".

Pigeons: Points in S, a bounded region of $\mathbb {R} ^{n}$ with total volume V > 1.

Pigeonholes: Positions in unit 'cube' (or equivalent of unit cube in $\mathbb {R} ^{n}$ ).

Proof of Minkowski's Theorem

Exercises

Exercises:

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