Given a metric space $Y$ with $y\in Y$, $r>0$ and $\epsilon>0$, we say that $y$ is $(k,\epsilon,r)$-symmetric if there exists a $k$-symmetric space $Y^{\prime}$ such that $d_{GH}(B_{r}(y),B_{r}(y^{\prime}))<\epsilon r$, where $y^{\prime}\in Y^{\prime}$ is a vertex.