  We construct for every finite-dimensional Alexandrov space $A$ and every
point $p \in A$ a $2$-convex function $f_p$ in a small neighborhood around $p$,
which approximates $\operatorname{dist}_p^2$ up to second order. Moreover, the
function $f_p$ can be lifted to Gromov-Hausdorff close Alexandrov spaces of the
same dimension.
