Geometry of numbers

In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by Hermann Minkowski (1910).

The geometry of numbers has a close relationship with other fields of mathematics, especially functional analysis and Diophantine approximation, the problem of finding rational numbers that approximate an irrational quantity.

Suppose that Γ is a lattice in n-dimensional Euclidean space Rⁿ and K is a convex centrally symmetric body. Minkowski's theorem, sometimes called Minkowski's first theorem, states that if ${\displaystyle \mathrm {vol} (K)>2^{n}\mathrm {vol} (\mathbb {R} ^{n}/\Gamma )}$, then K contains a nonzero vector in Γ.

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