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# Questions tagged [mg.metric-geometry]

Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.

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### Can planar set contain even many vertices of every unit equilateral triangle?

Is there a nonempty planar set that contains $0$ or $2$ vertices from each unit equilateral triangle? I know that such a set cannot be measurable. In fact, my motivation is to extend a Falconer-Croft ...
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### Pointed version of Perelman stability theorem

I am wondering if there is a version of the Perelman stability theorem which says approximately the following: Let $\{(X_i,p_i)\}$ be a sequence of pointed $n$-dimensional complete Alexandrov spaces ...
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If $D_r = \{v\in \mathbb{R}^2 : 0 \lt |v| \lt r\}$, consider the map $f_r: D_r \to D_r$ given by: $$f_r(x,y) = \frac{\sqrt{r^2-x^2-y^2}}{\sqrt{x^2+y^2}}\left(-y,x\right)$$ Geometrically, $f_r(v) \... 0answers 42 views ### Bound on the distance from points to the boundary of a hyperbolic surface Fix$\epsilon\in\mathbb{R}_{>0}$,$\Sigma$a surface with boundary and let$\mathcal{T}_{\Sigma}(L_{1},...,L_{n})$denote the Teichmüller space of hyperbolic structures of$\Sigma$with geodesic ... 0answers 54 views ### Approximation of 2-dimensional Alexandrov spaces Consider a closed 2-dimensional surface (not necessarily orientable) with a metric with curvature at least -1 in the sense of Alexandrov. It it true that on this surface there is a sequence of ... 1answer 133 views ### Geodesic preserving diffeomorphisms of constant curvature spaces Let$X$be either Euclidean space$\mathbb{R}^n$, the sphere$\mathbb{S}^n$, or hyperbolic space$\mathbb{H}^n$. I would like to have a classification of all diffeomorphisms$X\to X$which map ... 1answer 76 views ### A.D. Alexandrov imbedding theorem for metrics with symmetry A well known theorem due to A.D. Alexandrov says that any metric on the 2-sphere$S^2\$ with curvature at least -1 (in the sense of Alexandrov) can be isometrically realized either as convex surface in ...

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