# Measure of the boundary of Alexandrov space

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.)

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 $$\partial X$$.