Minkowski's theorem (geometry of numbers)

Minkowski's theorem states that a centrally symmetric convex body in Euclidean space contains a nonzero point of a lattice if its volume exceeds a bound set by the lattice's covolume.

Lattice: A discrete subgroup of Euclidean space generated by linearly independent vectors. The theorem measures its threshold using the volume of a lattice's fundamental region.
Minkowski's convex body theorem: The standard formulation guaranteeing a nonzero lattice point when a centrally symmetric convex body has volume greater than 2ⁿ times the lattice covolume. This is the theorem's commonly used form, with its dimension-dependent constant made explicit.
Determinant: A scalar associated with a square matrix that measures signed volume scaling. A basis matrix's determinant gives the lattice covolume used in the theorem.
Hermann Minkowski: A German mathematician who developed the geometry of numbers and introduced spacetime geometry. He proved the theorem that bears his name and shaped its mathematical setting.
Covolume: The volume of a fundamental region of a lattice in its ambient space. Minkowski's volume bound is expressed as a multiple of this quantity.
Minkowski's first theorem: A theorem bounding the length of a shortest nonzero vector in a lattice using its determinant and a convex body. It applies the convex-body principle to produce short lattice vectors.
Geometry of numbers: The study of arithmetic questions through the geometry of lattices and convex bodies. Minkowski's theorem is a central bridge between this field's geometric and arithmetic methods.
Number theory: The branch of mathematics concerned with integers and their properties. The theorem applies geometric reasoning to questions about integer solutions and approximations.
Convex set: A set containing every line segment between any two of its points. Convexity is one of the geometric conditions that makes the lattice-point guarantee hold.
Minkowski's second theorem: A theorem relating a lattice's determinant to the product of successive minima of a convex body. It extends the single-point guarantee to bounds on several independent lattice directions.

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