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.