title: Hermite constant
text: # Hermite constant
  
  In mathematics, the Hermite constant, named after Charles Hermite, determines how long a shortest element of a lattice in Euclidean space can be.
  
  The constant $γ_n$ for integers n > 0 is defined as follows. For a lattice L in Euclidean space $R^n$ with unit covolume, i.e. $vol(R^n/L)$ = 1, let $λ_1(L)$ denote the least length of a nonzero element of L. Then $√γ_n$ is the maximum of $λ_1(L)$ over all such lattices L.
  
  The square root in the definition of the Hermite constant is a matter of historical convention.
  
  Alternatively, the Hermite constant $γ_n$ can be defined as the square of the maximal systole of a flat n-dimensional torus of unit volume.
  
  ## Example
  
  The Hermite constant is known in dimensions 1–8 and 24.
  
  | n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 24 |
  | - | - | - | - | - | - | - | - | - | - |
  | $\gamma _{n}^{n}$ | $1$ | ${\frac {4}{3}}$ | $2$ | $4$ | $8$ | ${\frac {64}{3}}$ | $64$ | $2^{8}$ | $4^{24}$ |
  
  For n = 2, one has $γ_2$ = 2/√3. This value is attained by the hexagonal lattice of the Eisenstein integers.
  
  ## Estimates
  
  It is known that
  
      $\gamma _{n}\leq \left({\frac {4}{3}}\right)^{\frac {n-1}{2}}.$
  
  A stronger estimate due to Hans Frederick Blichfeldt is
  
      $\gamma _{n}\leq \left({\frac {2}{\pi }}\right)\Gamma \left(2+{\frac {n}{2}}\right)^{\frac {2}{n}},$
  
  where $\Gamma (x)$ is the gamma function.
  
  