Geometry of numbers

Minkowski's results

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 v o l ( K ) > 2 n v o l ( R n / Γ ) , then K contains a nonzero vector in Γ.

The successive minimum λ_k is defined to be the inf of the numbers λ such that λK contains k linearly independent vectors of Γ. Minkowski's theorem on successive minima, sometimes called Minkowski's second theorem, is a strengthening of his first theorem and states that

Later research in the geometry of numbers

In 1930-1960 research on the geometry of numbers was conducted by many number theorists (including Louis Mordell, Harold Davenport and Carl Ludwig Siegel). In recent years, Lenstra, Brion, and Barvinok have developed combinatorial theories that enumerate the lattice points in some convex bodies.

Subspace theorem of W. M. Schmidt

In the geometry of numbers, the subspace theorem was obtained by Wolfgang M. Schmidt in 1972. It states that if n is a positive integer, and L₁,...,L_n are linearly independent linear forms in n variables with algebraic coefficients and if ε>0 is any given real number, then the non-zero integer points x in n coordinates with

lie in a finite number of proper subspaces of Qⁿ.

Influence on functional analysis

Minkowski's geometry of numbers had a profound influence on functional analysis. Minkowski proved that symmetric convex bodies induce norms in finite-dimensional vector spaces. Minkowski's theorem was generalized to topological vector spaces by Kolmogorov, whose theorem states that the symmetric convex sets that are closed and bounded generate the topology of a Banach space.

Researchers continue to study generalizations to star-shaped sets and other non-convex sets.