Hermite Constant for Dimension 2

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The Hermite constant for dimension $2$ is:

$\gamma_2 = \dfrac 2 {\sqrt 3}$

or, as it is often presented:

$\paren {\gamma_2}^2 = \dfrac 4 3$


The statement of the result to be proved can be expressed as:

There exist non-zero $x$ and $y$ such that:
$\paren {a x^2 + 2 b x y + c y^2}^2 \le \size {a c - b^2} \times \dfrac 4 3$