# Questions tagged [mg.metric-geometry]

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

2,339 questions
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### Computing the volume of intersection between a hyper-rectangle and a ball

$C$ is a region bounded above, below coordinate-wise by $\overline{c},\ \underline{c}\in [-1,1]^d.$ Is there a procedure not exponentially complex with $d$ that computes the volume of $C\cap B(0,1)$ ...
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### Bound on change in relative length from 'well-behaved' Jacobian?

(This question was originally asked on Mathematics Stack Exchange, and sat there for several weeks with low views and no answers.) Let $\phi$ and $\gamma$ be rectifiable curves in the same length ...
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### n-D Gauss circle problem over a rectangle

I would like to approximate the amount of points in $\left(2^{-a\cdot n}\mathbb{Z}^n\right)\cap B^n(0,1)\cap C^n$ where $a>0,\ B^n(0,1)$ is the unit nball and $C^n$ is some rectangular domain ...
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### Is volume of abstract polytope realisation bounded by length of edges?

Suppose we have abstract polytope F of dimension d ( that is the greatest rank facet has rank n). Such abstract object may have realisations in d-dimensional Euclidean space as polytopes $A_i(F)$, and ...
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### Geodesics between boundary points of a hyperbolic space

Let $X$ be a (not necessarily proper) hyperbolic space. Following Gromov, we define the boundary of $X$ as the set of equivalence classes of sequences convergent at infinity. In general, it is not ...
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### Canonical Metrics on 3- and 4-Manifolds

From the Uniformization Theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ boundaries/punctures, subject to the condition $2g+n\ge 3$, contains a unique ...
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### Relative to Isoperimetric inequality with n-polygon

Since Brahmagupta's formula and Bretschneider's formula we have the inequality: Any two quardrilaterals $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ with the same sidelengths and $A_1A_2A_3A_4$ is a cyclic ...
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### Filling points to a simplex in models for EG

I have a question which is related to higher Dehn functions of groups. I also have a group $G$ with a finite $K(G,1)$. Let us denote by $EG$ the universal cover of this complex. We choose a path-...
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### Strengthened version of Isoperimetric inequality with n-polygon

Let $ABCD$ be a convex quadrilateral with the lengths $a, b, c, d$ and the area $S$. The main result in our paper equivalent to: a^2+b^2+c^2+d^2 \ge 4S + \frac{\sqrt{3}-1}{\sqrt{3}}\...
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### Tangled random triangles: One giant component?

Suppose you have $n$ triangles whose corners are random points on a sphere $S$ in $\mathbb{R}^3$. Viewing the triangles as built from rigid bars as edges, two triangles are linked if they cannot be ...
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### Odds on rolling a rhombicosidodecahedron

This is more of a curiosity to me, but I'm sure I don't have the mathematical skills to answer it. That said... I took a look at several other posts with questions that relate to this one, but I ...
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### Wasserstein distance to the set of Gaussians ; Boltzman dissipation rate

I am interested in the $2$-Wasserstein distance for probabilities over ${\mathbb R}^n$, $$W_2(\mu,\nu)=\left(\inf\int_{{\mathbb R}^n\times{\mathbb R}^n}|w-v|^2\pi(v,w)\right)^{1/2}$$ where the infimum ...
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### Embedding the $\ 2^n+1$-point metric 1-space

A metric space $\ (S\ d)\$ is said to be a 1-space $\ \Leftarrow:\Rightarrow\ \forall_{x\ y\in S}\ (x\ne y\ \Rightarrow\ d(x\ y)=1).$ Question:   Do there exist a non-negative integer $n,\$ ...
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### Is each compact metric space a subset of a compact absolute 1-Lipschitz retract?

A metric space $X$ is called an absolute $L$-Lipschitz retract if for any metric space $Y$ containing $X$ there exists a Lipschitz retraction $r:Y\to X$ with Lipschitz constant $Lip(r)\le L$. ...
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### Doubling dimension vs other metric dimensions

For separable metric spaces, three fundamental notions of dimension are equivalent: $$\text{dim }X = \text{Ind }X = \text{ind }X ,$$ Where does the doubling dimension fit into the picture?
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### Sets of points avoiding small angles

(1) $\mathbb{R}^2$. I'd like to place $n$ points in the plane so that the smallest angle they determine is as large as possible. In a sense, such a point set is in very general position, not only ...
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### How does Siegel's Hilbert-Blumenthal fundamental domain differ from Götsky's?

This question has been changed to something related but different from the original question. Thanks to @paulgarrett for chatting with me and helping me hone in on a more interesting part. The first ...
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### Is there any way to deform a Riemannian metric while keeping a positive curvature lower bound?

This is a somehow general question. Suppose you have a Riemannian manifold with sectional curvature lower bound, say $\kappa$. We want to find a way to deform the metric in a way such that the lower ...
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### Smooth approximation of locally CAT(-1) metrics

It is a well known fact that locally CAT(-1) metrics on surfaces can be approximated by hyperbolic polyhedral metrics with cone singularities: roughly speaking you pick a geodesic triangulation of the ...
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### Mathematical theory of aesthetics

The notion of beauty has historically led many mathematicians to fruitful work. Yet, I have yet to find a mathematical text which has attempted to elucidate what exactly makes certain geometric ...
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### Local isometry of complete length spaces that is not a covering map

Let $\pi:\widetilde{M}\to M$ be a surjective local isometry between complete length spaces (local isometry means that every point $x\in \widetilde{M}$ has a neighborhood which is isometrically mapped ...
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### Bruhat-Tits building of $SL_n(\mathbb{Q})$, hyperbolic isometries and its axis

Consider $G=SL_n(\mathbb{Q})$ and $p$ a prime integer. Associated to $G$ and $p$ we have its Bruhat-Tits building $\Delta$. It is well known that $\Delta$ can be provided with a canonical $CAT(0)$ ...