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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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      1answer
      84 views

      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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      0answers
      49 views

      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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      0answers
      81 views

      Move one element of finite set out from A in plane

      Suppose we are given two sets, $S$ and $A$ in the plane, such that $S$ is finite, with a special point, $s_0$, while neither $A$ nor its complement is a null-set, i.e., the outer Lebesgue measure of $...
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      46 views

      If all length metrics are strong equivalent on a closed connected topology manifold?

      Let $M$ be a connected closed topology manifold and $d$ is a length metric (or an inner metric) on it , i.e. the distance between every pair of points is equal to the infimum of the length of ...
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      84 views

      Why 2-torii with Gauss curvature $\geq -1$ cannot collapse to segment?

      Let $\{(\mathbb{T}^2,g_i)\}_{i=1}^\infty$ be a sequence of 2-dimensional torii with smooth Riemannian metrics with Gauss curvature at least -1. It was explained in the final answer to the post Gromov-...
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      1answer
      128 views

      Bending surfaces in Riemannian manifolds

      Let $S$ be an immersed surface in $\mathbb{R}^3$ (with the flat metric). We will call it flexible if there exists a smooth (or whatever regular) family of immersions $s_t: S\to \mathbb{R}^3$, such ...
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      141 views

      Vietoris-Rips complex and coarse geometry

      Let $K$ be an infinite countable subset of Euclidean space $E$ such any point of $E$ is within distance 1 of some point of $K$. In the language of John Roe's "coarse geometry", this implies that $K$ ...
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      38 views

      Trapping lightrays under nonstandard reflections and/or paths

      Almost every version of trapping lightrays with mirrors is either resolved---usually negatively---or open: "It is unknown whether one can construct a polygonal trap for a parallel beam of light": ...
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      56 views

      Is there any edge- but not vertex-transitive polytope in $d\ge 4$ dimensions?

      I consider convex polytopes $P\subset\Bbb R^d$. The polytope is called vertex- resp. edge-transitive, if any vertex resp. edge can be mapped to any other by a symmetry of the polytope. I am looking ...
      1
      vote
      1answer
      64 views

      Derivative of distance function to a closed, rectifiable set

      Let $\Gamma \subset \mathbf{R}^d$ be a closed, countably $n$-rectifiable set. Is there any reasonable way to write the derivatives $$ \frac{\partial}{\partial x_i} \mathrm{dist}\, (x,\Gamma) $$ for $x ...
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      0answers
      113 views

      Area-preserving map of punctured disk to itself

      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) \...
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      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 ...
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      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 ...
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      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 ...
      2
      votes
      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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