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We find the largest convex minorant of the function ¶ \begin{equation*} F\left( x,y\right) =ax^{2}+xy+by^{2} \end{equation*} ¶ where $a,b$ are positive constants and $x\geq 0,\ y\geq 0$. We explain how the problem is closely connected with\ finding the ground state Thomas-Fermi electron density for a spin\ polarized quantum mechanical system with the Fermi-Amaldi correction.