# Questions tagged [mg.metric-geometry]

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

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57 views

### Asymptotic bound on minimum epsilon cover of arbitrary manifolds

Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\varepsilon)$ denote the minimal cardinal of an $\varepsilon$-cover $P$ of $M$; ...

**-1**

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**1**answer

70 views

### Reference Request: Carnot Group Not Containing Group of Isometries

This question is a follow-up to this post, from which I quote:
Let $\mathfrak{e}$ be the 3-dimensional Lie algebra with basis $(H,X,Y)$ and bracket $[H,X]=Y$, $[H,Y]=-X$, $[X,Y]=0$. It is ...

**1**

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36 views

### Describing hull of vertex intersections of two convex bounded polytopes?

We have two convex bounded polytopes $P_1$ and $P_2$ where
a. $P_2\subseteq P_1$
b. $\mathcal{V}(P_2)\cap\mathcal{V}(P_1)\neq\emptyset$.
Is there a name for the polytope $P=\mbox{Conv}(\mathcal{V}(...

**5**

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**0**answers

109 views

### How to calculate the volume of a parallelepiped in a normed space?

Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a ...

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16 views

### Non-collapsed Alexandrov spaces, level surface of regular map homeo to its lifting?

Let $X_i$ be n dimensional, no boundary Alexandrov spaces with curvature $\geqslant -1$ and diameter $\leqslant D$. Suppose that $X_i$ converge to an n dimensional Alexandrov space $X$. Then by ...

**5**

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**0**answers

136 views

### Area of $n$-sphere contained outside $\ell_1$ ball

For a given $r>1$, what is the surface area of $\mathbb S^{n-1}$ (the sphere of radius 1 in $\mathbb R^n$) which is contained outside of the $\ell_1$ ball of radius $r$? Or equivalently, if $X\sim ...

**4**

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70 views

### Discrepancy of the finite approximation of the Lebesgue measure

Let $\mu$ be a probabilistic measure on the unit square $Q$ which is the average of $N$ delta-measures in some points in this square; let $\lambda$ denote the Lebesgue measure on $Q$. What is the rate ...

**7**

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**2**answers

471 views

### If a (distance) metric on a connected Riemannian manifold locally agrees with the Riemannian metric, is it equal to the induced metric?

Let $(M,g)$ be a connected Riemannian manifold.
Let $d_g$ be the induced distance metric of $g$. Now let $d$ be some other metric on $M$.
Suppose that for each $x \in M$, there is a neighborhood $U$ ...

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**0**answers

38 views

### Existence of $1$-Lipschitz map between triangles

Crosspost from math.SE
Consider two (Euclidean) triangles $T$ and $T'$.
Let's say that $T$ majorizes $T'$ if there exists a 1-Lipschitz map that sends
vertices to vertices and sides to sides (for ...

**5**

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**3**answers

436 views

### Alexandrov's generalization of Cauchy's rigidity theorem

Wikipedia states that A. D. Alexandrov generalized Cauchy's rigidity theorem for polyhedra to higher dimensions.
The relevant statement in the article is not linked to any source. The sources at the ...

**3**

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**0**answers

88 views

### Hyperbolic Intercept (Thales) Theorem

Is there an Intercept theorem (from Thales, but don't mistake it with the Thales theorem in a circle) in hyperbolic geometry?
Euclidean Intercept Theorem:
Let S,A,B,C,D be 5 points, such that SA, SC, ...

**8**

votes

**1**answer

475 views

### Illustrating that universal optimality is stronger than sphere packing

I'm a physicist interested in the conformal bootstrap, one version of which was recently shown to have many similarities to the problem of sphere packing. Sphere packing in $\mathbf{R}^d$ has been ...

**1**

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**1**answer

118 views

### Does fractallity depend on the Riemannian metric?

Edit: According to comment of Andre Henriques we revise the question:
In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that ...

**2**

votes

**1**answer

67 views

### The space of complex structure compatible with metric

Why the space of all complex structure on a $2n$-dimensional vector space which compatible with a positive definite metric is diffeomorphic to $ O(2n)/U(n) $ ?

**10**

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**1**answer

290 views

### Growing a chain of unit-area triangles: Fills the plane?

Define a process to start with a unit-area equilateral triangle,
and at each step glue on another unit-area triangle.
$50$ ...