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Let $X$ be a compact $n$-dimensional Alexandrov space with curvature bounded below. Let $\partial X$ denote its boundary in the sense of the theory of Alexandrov spaces.

Is it true that if $\partial X\ne \emptyset$ then it has finite and positive $(n-1)$-Hausdorff measure? (The case $n=2$ is already interesting to me.)

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It can be proved by induction on $n$. Base case $n=1$. The step follows since gradient exponent is locally Lipschitz and it maps $T_p(\partial X)=\mathrm{Cone}[\Sigma_p(\partial X)]$ to a neighborhood of $p$ in $\partial X$.

The statement also holds for extremal subsets of arbitrary dimension.

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