# Questions tagged [mg.metric-geometry]

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

2,339 questions
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### Is the conjecture true for n-sphere $(n>2)$? [closed]

This is higher dimension conjecture of Problem 3845 in Crux Mathematicorum and Theorem 2 in here: PS: This figure is very nice, this is also generalization of Brianchon’s theorem, The Pascal theorem, ...
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### Measure of set of vectors whose outer product are bounded

Let consider the canonical Euclidean space $E = \mathbb{R}^n$, endowed with the Lebesgue measure $\mu$. Define the map $v_k: E^k \rightarrow \mathbb{R}$ that sends a $k$-uple $x_1,\cdots, x_k$ of ...
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### Inequality on simplex with circumscribed sphere

I'm looking for a proof for this problem on simplex which I think it is true Question. Let $\mathcal{A}=A_0A_1...A_n$ be a simplex in $\Bbb E^n$. $(S)$ is circumscribed sphere of $\mathcal{A}$ with ...
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### Centroid and center circumscribed spheres in simplex

I'm looking for a proof for this problem on simplex which I think it is true Question. $A_0A_1...A_n$ is a simplex in the Euclidean space $\Bbb E^n$. $G$ is its centroid and its center ...
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### Euler line in metric space [closed]

If the answer about my question here is "yes" Coplanar set in metric space. We shall have the concept of four coplanar points in metric space $(\Bbb M,d).$ I propose an idea about Euler line in ...
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### A cube is placed inside another cube

I known following problem with two square $$Area(1)+Area(3)=Area(2)+Area(4).$$ My question. Is this problem true for two cubes? We place a cube $XYZT.X'Y'Z'T$ into another cube $ABCD.A'B'C'D',$ ...
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### Distance between quadric surface and point or Intersection of sphere and quadric surface

I asked a similar question on math.stackexchange, but the answer wasn't quite ideal for my application. Apparently analytic solutions are surprisingly rare for general quadric distances. Given a ...
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### $N$-$th$ closed chain of six circles

Since 2013, I found a very nice configuration: $N$-th closed chain of six circles. This is a generalization of theorem 1, problem 2 in here and theorem 2 in here and here (and is also generalization ...
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### Bi-Lipschitz extension

Given a bi-Lipschitz homeomorphism $\Phi:\mathbb{B}^n(0,1)\to\mathbb{R}^n$, (that is a bi-Lipschitz map onto the image), can one find a bi-Lipschitz homeomorphism $\Psi:\mathbb{R}^n\to\mathbb{R}^n$ ...
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### Pascal theorem for three dimensions

A year ago I found the Pascal theorem for three dimentions as follows: Let $(C_1)$, $(C_2)$ be two conics on the same Ellipsoid, (or Hyperboloid, or Paraboloid). Let $A_1$, $A_2$, $A_3$, $A_4$, $A_5$,...
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### Wasserstein distance to the set of Gaussians, relation to Boltzman dissipation rate

$\newcommand{\RR}{\mathbb{R}}$ I am interested in the 2-Wasserstein distance for probabilities over $\RR^n$, $W_2(μ,ν)=\left(\inf\int_{\RR^n×\RR^n}|w−v|^2dπ(v,w)\right)^1/2$ where the infimum is taken ...
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### Fitting one Polygon in another

I have two Polygons A and B and I want to find the position, rotation and scale of B, so it fits into A and has the maximum Area possible. Also both can be concave. I did some research but couldn't ...
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### Divergence of Groups and Metric Spaces

Several papers, including this and this claim that divergence of finitely generated groups and metric spaces have been introduced by Misha Gromov in his paper "Asymptotic invariants of infinite groups"...
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### An ellipse through 12 points related to Golden ratio

I am looking for a proof of the problem as follows: Let $ABC$ be a triangle, let points $D$, $E$ be chosen on $BC$, points $F$, $G$ be chosen on $CA$, points $H$, $I$ be chosen on $AB$, such that $IF$...
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### If a triangle can be displaced without distortion, must the surface have constant curvature?

Suppose $S$ is a Riemannian 2-manifold (e.g. a surface in $\mathbb{R}^3$). Let $T$ be a geodesic triangle on $S$: a triangle whose edges are geodesics. If $T$ can be moved around arbitrarily on $S$ ...
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### Regularity of geodesics

If $M$ is a graph of a $C^1$ function $f:\mathbb{R}^n\to\mathbb{R}$, is it true that the length minimizing geodesics on $M$ are $C^1$? I expect a counterexample. For a related discussion see Metric ...
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### In 3D point groups, does $[\Gamma_{e}\otimes\Gamma_e] = \Gamma_{Rot_z} \forall$ degenerate $\Gamma_e$ hold in general?

In the following I am referring to groups exclusively describing 3D point symmetries. I use the Schönflies notation for groups and their elements and the Mulliken symbols to describe their irreducible ...
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### Tying knots via gravity-assisted spaceship trajectories

Q. Can every knot be realized as the trajectory of a spaceship weaving among a finite number of fixed planets, subject to gravity alone?           To make this more ...
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### What is the name of the 65537-gon? [closed]

I know the name of the heptadecagon (17 sides) and the diacosipentacontaheptagon (257 sides). But what is the name of the polygon with 65537 sides? I am unable to figure it.
What do we know about the covering number of $L$-Lipschitz functions mapping say, $\mathbb{R}^n \rightarrow \mathbb{R}$ for some $L >0$? Only 2 results I have found so far are, That the $\infty$-...
Let $Q,Q'$ be two planar polygons with the same number $n>3$ of vertices. There is a correspondence between vertices of $Q$ and $Q'$: to any vertex $z$ of $Q$ corresponds a unique vertex $z'$ of \$Q'...