Upper Bound of Hermite Constant

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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



Sources

  • 1929: H.F. BlichfeldtThe minimum value of quadratic forms, and the closest packing of spheres (Math. Ann. Vol. 101: pp. 605 – 608)