TITLE: Continuity of volume of boundary of Riemannian manifolds in the Gromov-Hausdorff sense
QUESTION [6 upvotes]: Let $\{X_i^n\}$ be a sequence of smooth compact Riemannian $n$-dimensional manifolds with boundary. Assume that this sequence has uniformy bounded below sectional curvature, and each $X_i$ is geodesically locally convex near its boundary (the latter assumption is equivalent that $X_i$ is an Alexandrov space). Assume $\{X_i\}$ converges in the Gromov-Hausdorff sense to a compact smooth Riemannian manifold $X$ with boundary and $\dim X=n$, i.e. there is no collapse.

Is it true that volume of the bondary $vol_{n-1}(\partial X_i)$ converges to $vol_{n-1}(\partial X)$? The case $n=2$ is already interesting to me.

Remarks. (1) By the Perelman stability theorem $X_i$ is homeomorphic to $X$ for $i\gg 1$. 
(2) By Burago-Gromov-Perelman one has convergence of volumes of manifolds $vol_n(X_i)\to vol_n(X)$.

REPLY [3 votes]: [The statement holds for a sequence of extremal susbsets (not necessary boundary).]
According to Theorem 1.2. in my "Applications of quasigeodesics and gradient curves",
$\partial X_i\to \partial X$ as length spaces in the sense of Gromov--Hausdorff.
Then the same argument as in Burago--Gromov--Perelman shows that $\mathrm{vol}\,\partial X_i\to \mathrm{vol}\,\partial X$.