# Questions tagged [mg.metric-geometry]

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

2,469
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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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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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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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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 ...

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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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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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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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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 ...

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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 ...