Toward canonical convex functions in Alexandrov spaces

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 ${\mathrm{dist}}_p^2$ up to second order. Moreover, the function $f_p$ can be lifted to Gromov-Hausdorff close Alexandrov spaces of the same dimension.