# Questions tagged [alexandrov-geometry]

Alexandrov geometry studies non smooth analogues of Riemannian manifolds with curvature bounded from below or above. It includes spaces with curvature bounded below (briefly $\mathrm{CBB}[\kappa]$) and spaces with curvature bounded above (briefly $\mathrm{CAT}[\kappa]$).

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### Concavity of the distance function to the boundary of Alexandrov space

I was told that the following fact is true. Let $X$ be a finite dimensional Alexandrov space with non-negative curvature. Then the function $$x\mapsto dist(x, \partial X)$$ is concave (namely its ...
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### Estimate of volume of a ball on the boundary of Riemannian manifold

Let $M^n$ be a smooth compact Riemannian manifold with geodesically locally convex boundary and sectional curvature at least $-1$. Let $x\in M$ and $\varepsilon\in (0,1)$. Does there exist a ...
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### A Lipschitz limit of Riemannian metrics with curvature in $[-1,1]$

Let $(M,g)$ be a compact manifold with a metric $g$ (not necessarily Riemannian one). Suppose that $(M,g)$ is a Lipschitz limit of a sequence of Riemannian manifolds $(M_n,g_n)$ with the sectional ...
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### When is the quotient of a geodesic space again a geodesic space?

I asked this on the math.stackexchange forum about a week ago but did not get any answers so I figured I might try it here as well. This is a straight up copy paste from my question here. I am ...
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### Continuity of volume of boundary of Riemannian manifolds in the Gromov-Hausdorff sense

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 ...
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### Angle estimate in Alexandrov spaces

I am not sure that this is a research level question. Remark 10.9.4 in the book "A course in metric geometry" by Burago, Burago, Ivanov claims the following. Let $X$ be a finite dimensional ...
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### Rademacher type theorem for Alexandrov spaces

The classical Rademacher theorem says that any Lipschitz function on a doman in $\mathbb{R}^n$ has the first derivative almost everywhere. I am wondering if this result can be generalized as follows. ...
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### Curvature $\geq-1$ but not $\geq1$

(Edited again) In the following, for brevity, I will say that $$X\ \ \mathrm{has}\ \ \kappa_{\mathrm{max}}=k$$ if $X$ is a compact ($n$-dimensional with $n\geq2$, with empty boundary) Aleksandrov ...
Assume $f\colon \mathbb{D}\to\mathbb{R}^2$ is a harmonic map and $x\notin f(\partial\mathbb{D})$. Is it true that $f^{-1}\{x\}$ is totally disconnected? I hope that the answer is yes. But actually I ...