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Let $X^n$ be an $n$-dimensional Alexandrov space with curvature bounded below. A point $x\in X$ is called regular if the space of directions $\Sigma_x$ is isometric to the standard sphere $S^{n-1}$.

QUESTION 1. Is it true that the set of regular points has full Hausdorff measure?

(Rmk: Theorem 10.9.13 in the Burago-Burago-Ivanov book claims a weaker property: this set is everywhere dense, and moreover is a countable intersection of open everywhere dense subsets.)

If the answer is yes, a reference would be helpful.

QUESTION 2. Let now $X^n$ be a convex hypersurface in the Euclidean space $\mathbb{R}^{n+1}$. Let $x\in X$ be a smooth point of $X$, i.e. there is a unique supporting hyperplane at $x$. Is it true that $x$ is regular in the above sense?

(Rmk: if this is the case then the set of regular points on convex hypersurface should have full Hausdorff measure since the set of smooth points has full measure.)

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"Yes" to both questions.

For the second, take the projection to the tangent plane and note that its bi-Lipschitz in a small neighborhood of $x$ with constants as close to 1 as you want.

For the first one, see in 10.6 in "Alekandrov's Space with Curvature bounded from below" by Burago, Gromov and Perelman.

[In fact you can say bit more about regular set; it is convex and the complement is countably $(n-1)$-rectifiable; that is it lies in the images of countable collection of Lipschitz maps $\mathbb {R}^{n-1}\to X^n$. Moreover if there is no boundary then it is is countably $(n-2)$-rectifiable. One can say yet more --- in some sense all you know about singularities of convex surfaces is known for Alexandrov spaces.]

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  • $\begingroup$ Thanks very much for exhaustive answer! Your last sentence intrigued me. For convex functions there is Alexandrov's theorem that such a function almost everywhere has second derivative. Does it generalize to Alexandrov spaces? E.g. can one define the Riemann curvature tensor almost everywhere? (Should I formulate this in a separate question?) $\endgroup$ – MKO Aug 30 '15 at 15:22
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    $\begingroup$ @sva, yes, semiconcave functions on Alexandrov space have well defined Hessian almost everywhere (4.4 in math.psu.edu/petrunin/papers/alexandrov/Cstructure.pdf). $\endgroup$ – Anton Petrunin Aug 30 '15 at 15:31

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