Upper Bound of Hermite Constant

Theorem

Let $\gamma_n$ be the Hermite constant of dimension $n$.

Then:

$\gamma_n \le \dfrac {\paren {1 + \epsilon_n} n} {\pi e}$

where $e_n \to 0$


Proof


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Sources

  • 1929: H.F. Blichfeldt: The minimum value of quadratic forms, and the closest packing of spheres (Math. Ann. Vol. 101: pp. 605 – 608)
  • 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1,33333 33333 33 \ldots$
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