  In \cite{CC1}, Cheeger-Colding considered manifolds with lower Ricci
curvature bound and gave some almost rigidity results about warped products
including almost metric cone rigidity and quantitative splitting theorem. As a
generalization of manifolds with lower Ricci curvature bound, for metric
measure spaces in $RCD(K, N)$, $1<N<\infty$, splitting theorem \cite{Gi13} and
"volume cone implies metric cone" rigidity for balls and annuluses of a point
\cite{PG} have been proved. In this paper we will generalize Cheeger-Colding's
\cite{CC1} result about "almost volume cone implies almost metric cone for
annuluses of a compact subset " to $RCD(K, N)$-spaces. More precisely, consider
a $RCD(K, N)$-space $(X, d, \mathfrak m)$ and a Borel subset $\Omega\subset X$.
If the closed subset $S=\partial \Omega$ has finite outer curvature, the
diameter ${diam}(S)\leq D$ and the mean curvature of $S$ satisfies $$m(x)\leq
m, \, \forall x\in S,$$ and \begin{equation*}\mathfrak m(A_{a, b}(S))\geq
(1-\epsilon)\int_a^b \left({sn}'_H(r)+ \frac{m}{n-1}{sn}_H(r)\right)^{n-1}dr
\mathfrak m_S(S)\end{equation*} then $A_{a', b'}(S)$ is measured
Gromov-Hausdorff close to a warped product $(a', b')\times_{{sn}'_H(r)+
\frac{m}{n-1}{sn}_H(r)}Y,$ $A_{a, b}(S)=\{x\in X\setminus \Omega, \, a<d(x,
S)<b\}$, $a<a'<b'<b$, $Y$ is a metric space with finite components with each
component is a $RCD(0, N-1)$-space when $m=0, K=0$ and is a $RCD(N-2,
N-1)$-space for other cases and $H=\frac{K}{N-1}$. Note that when $m=0, K=0$,
our result is a kind of quantitative splitting theorem and in other cases it is
an almost metric cone rigidity.
  To prove this result, different from \cite{Gi13, PG}, we will use
\cite{GiT}'s second order differentiation formula and a method similar as
\cite{CC1}.
