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.



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> 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.
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**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 $vol\_n(X\_i)\to vol\_n(X)$.



Continuity of volume of boundary of Riemannian manifolds in the Gromov-Hausdorff sense
[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 $\mathrm{vol}\,\partial X\_i\to \mathrm{vol}\,\partial X$.


