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.)


1 Answer 1


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 $\partial X$. It proves that $\partial X$ has finite $(n-1)$-Hausdorff measure.

The differential of gradient exponent at the origin is the identity map. It follows that a small cube $\square$ around a regular point in $T_p(\partial X)$ is mapped amost isometrically; in particular the opposite faces of $\square$ go to sets on positive distance. By Besicovitch inequality, the image of $\square$ has positive $(n-1)$-Hausdorff measure.

The same proof shows that an $m$-dimensional extremal subset in a compact Alexandrov space has finite positive $m$-dimesnional Hausdoeff measure.

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    $\begingroup$ It seems to me that this argument only shows $dim_H \partial X \le n-1$. How can one prove $H_{n-1}(\partial X)>0$? $\endgroup$
    – Totoro
    Jun 7, 2021 at 10:32
  • $\begingroup$ @Totoro, it is fixed now, thank you. $\endgroup$ Jun 7, 2021 at 23:04
  • $\begingroup$ May I ask what is the definition of a regular point in $T_p(\partial X)$? As $\partial X$ may not be an Alexandrov space, the definition is unclear to me. Anyway, one can use the same method and prove by induction that any $m$-dimensional extremal set contains a "cube" which is the image of $I^m$ such that the opposite faces of the cube have positive distance. $\endgroup$
    – Totoro
    Jun 8, 2021 at 2:25
  • $\begingroup$ @Totoro A point $p$ is a regular point of subset S if the tangent space $T_pS$ is Euclidean. $\endgroup$ Jun 8, 2021 at 2:35
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    $\begingroup$ @AntonPetrunin "Regular points of extremal subsets in Alexandrov spaces" by Fujioka is a reference. Not sure if there are any earlier reference. $\endgroup$
    – Adterram
    Jun 9, 2021 at 1:28

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