Hermite constant
E502193
The Hermite constant is a number in each dimension that measures the densest possible lattice sphere packing, playing a central role in the geometry of numbers and lattice theory.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
invariant in the geometry of numbers
ⓘ
mathematical constant ⓘ |
| achievedBy |
D_4 root lattice in dimension 4
ⓘ
E_8 root lattice in dimension 8 ⓘ Leech lattice in dimension 24 NERFINISHED ⓘ face-centered cubic lattice in dimension 3 ⓘ hexagonal lattice in dimension 2 ⓘ |
| appearsIn |
Hermite–Minkowski theorem
NERFINISHED
ⓘ
bounds for discriminants of number fields ⓘ |
| classificationStatus | exact values known only for finitely many dimensions ⓘ |
| codomain | positive real numbers ⓘ |
| definitionUses |
determinant of a lattice
ⓘ
shortest nonzero lattice vector ⓘ |
| dependsOn | dimension n ⓘ |
| describes | densest possible lattice sphere packing in dimension n ⓘ |
| domain | positive integers n ≥ 1 ⓘ |
| field |
geometry of numbers
ⓘ
lattice theory ⓘ |
| introducedInContextOf | reduction theory of positive definite quadratic forms ⓘ |
| isSequence | {γ_n}_n ⓘ |
| lowerBoundGrowth | Ω(n / log n) ⓘ |
| monotonicity | nondecreasing in n ⓘ |
| namedAfter | Charles Hermite NERFINISHED ⓘ |
| relatedTo |
E_8 lattice
NERFINISHED
ⓘ
Leech lattice NERFINISHED ⓘ Minkowski’s theorem NERFINISHED ⓘ covering radius of lattices ⓘ kissing number problem ⓘ lattices in Euclidean space ⓘ root lattices ⓘ sphere packing density ⓘ successive minima of lattices ⓘ |
| researchStatus | subject of ongoing research in high dimensions ⓘ |
| symbol | γ_n ⓘ |
| upperBoundGrowth | O(n) ⓘ |
| usedFor |
Minkowski-type inequalities
NERFINISHED
ⓘ
bounding minima of quadratic forms ⓘ studying densest lattice sphere packings ⓘ transference theorems in geometry of numbers ⓘ |
| usedIn |
coding theory via lattice codes
ⓘ
cryptography based on lattices ⓘ |
| γ_1 | 1 ⓘ |
| γ_2 | 2/√3 ⓘ |
| γ_24 | 4 ⓘ |
| γ_3 | 2^{1/3} ⓘ |
| γ_4 | 2^{1/2} ⓘ |
| γ_8 | 2 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.