say that $h$ is $(\kappa,C,\delta,(p_l)_{l=1,\ldots,n-1})$-steep on $G'$ and, if all the $p_i=p$, we say that $h$ is $(\kappa,C,\delta,p)$-steep on $G'$.

We have the following proposition, which states that if a smooth function has a stably steep Taylor polynomial at some point, then the function is steep on a neighborhood of this point.

\[propsteep1\] Let $h : B_{\bar{r}} \rightarrow {\mathbb{R}}$ be a function of class $C^{m_0+1}$ such that $||\nabla h(0)||\geq\varpi>0$ and such that $T_0^{m_0}h$ is $(\rho',C',\delta')$-stably steep. Then for $r>0$ sufficiently small with respect to $\bar{r}, ||h||_{C^{m_0+1}(B_{\bar{r}})}, \rho', \varpi, m_0, C'$ and $\delta'$, the function $h$ is $(\kappa,C,\delta,m_0-1)$-steep on $B_{2r}$ with $$\kappa=\varpi/2, \quad C=C'/2, \quad \delta=r.$$

This proposition is analogous to Theorem 2.2 of [@BFN15]; however, the setting being slightly different we give the details.

We let $M=||h||_{C^{m_0+1}(B_{\bar{r}})}$, and we assume $3r<\bar{r}$. Then observe that for $r$ sufficiently small with respect to $M$ and $\varpi$, we have $$||\nabla h(I)||\geq \kappa=\varpi/2, \quad I \in B_{2r}.$$ Since $P_0:=T_0^{m_0}h$ is $(\rho',C',\delta')$-stably steep, for $r$ sufficiently small with respect to $M$ and $\rho'$, $P_I:=T_I^{m_0}h$ is $(\rho'/2,C',\delta')$-stably steep for any $I \in B_{2r}$. By definition, for any vector subspace $\Lambda \subseteq {\mathbb{R}}^n$ of dimension $1 \leq l \leq n-1$, letting $P_{I,\Lambda}$ be the restriction of $P_I$ 