In
mathematics, the Minkowski–Hlawka theorem is a result on the
lattice packing of
hypersphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, call ...
s in dimension ''n'' > 1. It states that there is a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
of dimension ''n'', such that the corresponding best packing of hyperspheres with centres at the
lattice point
In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice poi ...
s has
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
Δ satisfying
:
with ζ the
Riemann zeta function. Here as ''n'' → ∞, ζ(''n'') → 1. The proof of this theorem is indirect and does not give an explicit example, however, and there is still no known simple and explicit way to construct lattices with packing densities exceeding this bound for arbitrary ''n''. In principle one can find explicit examples: for example, even just picking a few "random" lattices will work with high probability. The problem is that testing these lattices to see if they are solutions requires finding their shortest vectors, and the number of cases to check grows very fast with the dimension, so this could take a very long time.
This result was stated without proof by and proved by . The result is related to a linear
lower bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an eleme ...
for the
Hermite constant.
Siegel's theorem
proved the following generalization of the Minkowski–Hlawka theorem. If ''S'' is a bounded set in R
''n'' with Jordan volume vol(''S'') then the average number of nonzero lattice vectors in ''S'' is vol(''S'')/''D'', where the average is taken over all lattices with a fundamental domain of volume ''D'', and similarly the average number of primitive lattice vectors in ''S'' is vol(''S'')/''D''ζ(''n'').
The Minkowski–Hlawka theorem follows easily from this, using the fact that if ''S'' is a star-shaped centrally symmetric body (such as a ball) containing less than 2 primitive lattice vectors then it contains no nonzero lattice vectors.
See also
*
Kepler conjecture
The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling s ...
References
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{{DEFAULTSORT:Minkowski-Hlawka theorem
Geometry of numbers
Theorems in geometry
Hermann Minkowski