Hermite constant

E502193

invariant in the geometry of numbers mathematical constant

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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (3)

Label	Occurrences
Hermite constant canonical	1
Hermite constant bounds	1
Hermite’s constant	1

How this entity was disambiguated

This entity first appeared as the object of triple T5191867 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Hermite constant
Context triple: [Charles Hermite, hasConceptNamedAfter, Hermite constant]

A. Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
B. Khinchin–Lévy constant
The Khinchin–Lévy constant is a mathematical constant arising in metric number theory and continued fractions, describing the typical exponential growth rate of the denominators of convergents for almost all real numbers.
C. Khinchin's constant
Khinchin's constant is a mathematical constant that arises in metric number theory, describing the almost-sure geometric mean of the partial quotients in the continued fraction expansions of real numbers.
D. Waring's problem
Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.
E. Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Hermite constant
Target entity description: 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.

A. Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
B. Khinchin–Lévy constant
The Khinchin–Lévy constant is a mathematical constant arising in metric number theory and continued fractions, describing the typical exponential growth rate of the denominators of convergents for almost all real numbers.
C. Khinchin's constant
Khinchin's constant is a mathematical constant that arises in metric number theory, describing the almost-sure geometric mean of the partial quotients in the continued fraction expansions of real numbers.
D. Waring's problem
Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.
E. Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
F. None of above. chosen

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 ⓘ linked to: Leech lattice face-centered cubic lattice in dimension 3 ⓘ hexagonal lattice in dimension 2 ⓘ
appearsIn	Hermite–Minkowski theorem ⓘ 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 ⓘ
relatedTo	E_8 lattice ⓘ linked to: E8 lattice Leech lattice ⓘ Minkowski’s theorem ⓘ linked to: Minkowski’s theorem on convex sets 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 ⓘ linked to: Minkowski inequality 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 ⓘ

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Charles Hermite → hasConceptNamedAfter → Hermite constant ⓘ

Hermite–Minkowski theorem → relatedTo → Hermite’s constant ⓘ

linked to: Hermite constant

geometry of numbers → hasTheorem → Hermite constant bounds ⓘ