# Questions tagged [mg.metric-geometry]

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

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### Reference request: Oldest (non-analytic) geometry books with (unsolved) exercises?

Per the title, what are some of the oldest (non-analytic) geometry books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there.

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### Sphere packing in a sphere

Let $S_a^d$ be the $(d-1)$-dimensional sphere of radius $a$ in $\mathbb{R}^d$. Let $r>0$ be a constant and $R=\nu r$ where $\nu>1$ (some constant). Are there any known upper bounds on the number ...

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**1**answer

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### How to compute the parameters of circumscribed hypershpere? [on hold]

Assume I have an $n$-dimensional simplex on the points $x_0, ..., x_n$ where each $x_i \in \mathbb{R}^n$. I would like to obtain the parameters (center and radius) of it's circumscribed n-dimensional ...

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### Metrically Ramsey ultrafilters

On Thuesday I was in Kyiv and discussed with Igor Protasov the system of MathOverflow and its power in answering mathematical problems. After this discussion Igor Protasov suggested to ask on MO the ...

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227 views

### Sums of squared distances between points on an $n$-sphere

I have discovered the following results about the sums of squared distances between points on an $n$-sphere (and proved them). To the best of my knowledge (and my advisor's knowledge), these results ...

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348 views

### Gluing hexagons to get a locally CAT(0) space

I believe that there are four ways to glue (all) the edges of a regular Euclidean hexagon to get a locally CAT(0) space:
The first two give the torus and the Klein bottle, respectively. What are the ...

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### Bishop-Gromov inequality strengthened for anisotropic metrics?

The Bishop-Gromov inequality provides an upper-bound on the rate of growth of volume of a ball of radius $r$ in spaces that have a lower-bound on the Ricci curvature, $Ric \geq (n-1)K$.
(I am ...

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205 views

### An isoperimetric inequality for curve in the plane?

Let $f(x,y)=0$ be a (smooth) simple closed curve $C$ on the plane and $R$ the region bounded by $C$ (appropriately oriented). Assume the origin lies in the interior of $R$.
QUESTION. Let $r=\sqrt{x^...

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### Polygons such that $n^2 $ times magnification of a polygon could be covered by exactly $n^2$ original polygon

While studying about covering problems in combinatorics, I got to a simple question:
What polygons can be covered exactly, without any area that is covered twice or area that is outside the covered ...

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**1**answer

531 views

### A quantitative version of Straszewicz's theorem?

Let $C$ be a compact convex subset of Euclidean space. Recall that $x\in C$ is an exposed point of $C$ if there is a plane $P$ such that $P\cap C = \{x\}$. It is obvious that exposed points are ...

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### Extensions of the Koebe–Andreev–Thurston theorem to sphere packing?

The Koebe–Andreev–Thurston theorem states that any planar graph can be represented
"in such a way that its vertices correspond to disjoint disks, which touch if and only if
the corresponding vertices ...

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**1**answer

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### Geodesics intersecting a submanifold

Let $U$ be an open subset in $R^n$ and let $N$ be a $C^1$-submanifold. We have a family of geodesics $\gamma:[0,1]\rightarrow U$ in U with respect to the euclidian metric. Each geodesic is ...

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### Uniform versus non-uniform group stability

Group stability considers the question of whether "almost"-homomorphisms are "close to" true homomorphisms. Here, "almost" and "close to" are made rigorous using a group metric.
More precisely, ...

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### Optimal packing of spheres tangent to a central sphere

Please consider a central, ordinary 2-sphere $S_1$, of some radius $r_1$, and a second ordinary sphere, $S_2$, of radius $r_2$, where $r_2 \leq r_1$.
My question concerns optimal values for the ...

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### Delaunay triangulations and convex hulls

This is a reference request.
I have the impression that those who work in computational geometry are accustomed to the following. You have some locally finite set of sites in $\mathbb{R}^n$ and you ...

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**1**answer

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### Any reference to an algorithm for finding the largest empty circle on a sphere (with great-circle distance)?

Given a set $S$ of 2D points in the plane, there are known algorithms for finding the largest empty circle ($LEC$) of the set of points.
The $LEC$ problem is stated in this way: find a $LEC$ whose ...

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**1**answer

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### Metric 1-current decomposition

I've been reading Paolini-Stepanov arcticle and in section 4, at page 6, they define a metric current from a transport:
$$T_{\eta}(\omega)=\int_{\Theta}[[\theta]](\omega)d\eta(\theta),$$
which ...

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### Isoperimetric-like inequality for non-connected sets

The classical isoperimetric inequality can be stated as follows: if $A$ and $B$ are sets in the plane with the same area, and if $B$ is a disk, then the perimeter of $A$ is larger than the perimeter ...

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77 views

### Uniformly Converging Metrization of Uniform Structure

This is related to trying to resolve the currently faulty second part of my answer to this question, but is by itself a purely real analysis question.
Let $X$ be a set with a uniform structure ...

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### Is a polytope with vertices on a sphere and all edges of same length already rigid?

Let's say $P\subset\Bbb R^d$ is some convex polytope with the following two properties:
all vertices are on a common sphere.
all edges are of the same length.
I suspect that such a polytope is ...

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### $L^{2}$ Betti number

Let $\tilde{X}$ be a non-compact oriented, Riemannian manifold adimits a smooth metric $\tilde{g}$ on which a discrete group $\Gamma$ of orientation-preserving isometrics acts freely so that the ...

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### Non-homeomorphic computable metric spaces whose computable points are computably homeomorphic

This is a follow-up of sorts to an earlier question on mine in that it should be easier to construct a positive example of this, if it exists.
To be clear about definitions, a computable metric space ...

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### Which cubic graphs can be orthogonally embedded in $\mathbb R^3$?

By an orthogonal embedding of a finite simple graph I mean an embedding in $\mathbb R^3$ such that each edge is parallel to one of the three axis. To avoid trivialities, let's require that (the ...

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### Uniformly sampling the solution space for points where the free termini of two rays, anchored at 3-space points, can intersect

I have two rays, one of length $L_1$ and one of length $L_2$. I anchor these rays, each at one end, on the 3-space points $p_1$ and $p_2$. Assuming that the Euclidean distance between $p_1$ and $p_2$...

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351 views

### Pointers/Papers on subdivision of planar quadrilateral meshes (PQ-Mesh) in 3D?

I'm interested in the subdivision of planar quadrilateral meshes (PQ-Meshes). Meshes consisting only of planar quadrilaterals, like discrete Voss surfaces and alike. I've been searching the web
for ...

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**1**answer

77 views

### Criterion for visuality of hyperbolic spaces

I am trying to understand the following sentence on p. 156 of Buyalo-Schroeder, Elements of asymptotic geometry: "Every cobounded, hyperbolic, proper, geodesic space is certainly visual."
Let $X$ be ...

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712 views

### Algorithm for k-medians in a convex polygon

Are there any known approximation algorithms or exact solution schemes for the k-medians problem in a convex polygon? That is, placing a collection of points $p_1,\dots,p_k \subset \mathbb{R}^2$ in a ...

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92 views

### Valid metric on a hyperbolic space

Note: originally posted on math.SE.
I'm looking at the distance that's defined in this paper on Poincaré Embeddings:
$d(\mathbf{u}, \mathbf{v}) = \operatorname{arccosh} \left(1 + 2\frac{\left\| \...

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### What is the meaning of Conjugate radius and Injectivity radius?

I review the text book of differential geometry and I find that the conjugate radius and injectivity radius are still enigmatic for me. Here is a quetion which I confuse it. I don't think it is true, ...

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373 views

### The Universality Theorem by Mnev for uniform oriented matroids of rank 4 and higher

According to the Universality Theorem by Mnev (see below theorem 8.6.6 from [1]), for any open semialgebraic variety V there is a uniform oriented matroid of rank 3 whose realization space is stably ...

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148 views

### Convex sets in Alexandrov spaces

Let $X$ be a compact Alexandrov space with $curv\geq 1$ (and without boundary). Does $X$ always have a nontrivial compact convex subset without boundary?
Definition of a convex subset: $A\subseteq X$ ...

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### Boundary surfaces in a 3d Voronoi tessellation with obstacles

Let $x_1,\dots,x_n$ be a set of points in $\mathbb{R}^3$ and let $\mathcal{O}_1 ,\dots, \mathcal{O}_m$ denote a set of polyhedral obstacles. What is the name for the surfaces that describe the ...

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### Term for a metric space for which the triangle inequality is strict?

Is there a standard term for a metric space in which $\rho(p,r) < \rho(p,q) + \rho(q,r)$ for any distinct $p$, $q$, $r$? Sort of the opposite of metric convexity.
For instance, a subset of ...

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### Can 4-space be partitioned into Klein bottles?

It is known that $\mathbb{R}^3$ can be partitioned into disjoint circles,
or into disjoint unit circles, or into congruent copies of a real-analytic curve
(Is it possible to partition $\mathbb R^3$ ...

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### A questions concerning Laguerre/Voronoi tessellations

Fix $n>1$ distinguished points $x_1,\ldots, x_n\in \mathbb R^d$, the Voronoi tessellations are the subsets $V_1,\ldots V_n\subset\mathbb R^d$ defined by
$$V_k~~ := ~~ \big\{x\in\mathbb R^d:\quad |...

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### Estimation of the Gromov–Wasserstein distance of spheres

Let $(X,d_X,\mu_X)$ and $(Y,d_Y,\mu_Y)$ be two metric measure spaces. A probability measure $\mu$ over $X\times Y$ is called a coupling if $(\pi_1)_\sharp \mu=\mu_X$ and $(\pi_2)_\sharp \mu=\mu_Y$. We ...

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447 views

### Disc bounded by a plane curve

Let $\Sigma$ be a sphere topologically embedded into $\mathbb{R}^3$.
Is it always possible to find a disc $\Delta\subset\Sigma$ which is bounded by a plane curve?
It is easy to find an open disc ...

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965 views

### Sofa in a snaky 3D corridor

What is the largest volume object that can pass though a
$1 \times 1 \times L$ "snaky" corridor, where $L$ is large
enough to be irrelvant, say $L > 6$.
...

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66 views

### Equal volume and projections

Given three unit vectors $u_1,u_2,u_3$ in $\mathbb{R}^3$, can we find some body $K \subset \mathbb{R}^3$ (probably convex) such that the following three things hold
(1) $|P_{u_1^\perp}K|=|P_{u_2^\...

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399 views

### Axiom of choice and a set in the plane that intersects every line in two points

In this question Subset of the plane that intersects every line exactly twice someone ask for a reference of a paper where they proof the result : ''There exist a subset of the plane that intersects ...

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### Probability of two Points being divided by an high-Dimensional Hyperplane

I have two points $x_1,x_2 \in \mathbb S^n $ which are distant $d$ from each other, where $d<<1$.
I also have a vector $v$ sampled uniformly at random from $\mathbb S^n$.
What is the ...

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### A special connected subset of the Cantor fan

Is there a dense connected subset $X$ of the Cantor fan $$(C\times [0,1])/(C\times \{1\})$$ such that for every two connected subsets $X_1,X_2\subseteq X$, the intersection $X_1\cap X_2$ is connected?
...

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### Locally compact metric spaces whose group of isometries generated by isometries of small displacement is transitive enough

I wasn't sure how to make the title any more precise than that.
Let $(X,d)$ be a locally compact metric space. For any $\varepsilon>0$ let $\mathrm{Aut}_\varepsilon (X)$ be the subgroup of the ...

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### Modulus of image of a curve family in a rectangle

I don't expect to get a positive answer to this question but I may as well try.
Let $R$ be the rectangle in $\mathbb{C}$ given by $\{z=x+iy: 0\leq x \leq l, 0 \leq y \leq h\}$ for some $l,h>0$. ...

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### Creating high quality figures of surfaces

I am not sure if this question is suitable for mo, it is more about visualization than math. Anyway, here it is:
What is the best way to visualize a 2-surface in Euclidean space with high quality?
...

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### Reference request: $\alpha$-Hölder spaces as double duals

If $(X,d)$ is a complete metric space, we define the $\alpha$-Hölder class $\Lambda_\alpha(X)$ as the subset of $C_b(X)$ satisfying that
$$
\sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^\alpha}.
$$
...

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**1**answer

47 views

### parametrize triangles meeting certain conditions

Consider triangles with angles alpha, beta, gamma such that gamma = 2 alpha, and sides (a,b,c) are integers. I want to parametrize such triangles by a single integer or rational variable.

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### Homeomorphism type of the horofunction boundary for nilpotent Lie groups

Consider a metric space $(X,d)$ and fix a base point $w$. A horofunction is a function of the form
$$\beta_y(x)=d(x,y)-d(w,y).$$
The map $y\mapsto \beta_y$ is an embedding of $X$ into the space of $1$-...

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931 views

### Placing points on a sphere so that no 3 lie close to the same plane

Motivation
I am working with arbitrary parallelopiped tilings given by projection from a higher dimensional space. The collection of tiles, and some properties of the higher dimensional space are ...

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400 views

### Important lines in triangle - reverse problem

It is known that if three numbers $x,y,z$ are the lengths of the edges of some triangle, then there exists a triangle with medians of length $x,y,z$. Also, if $x,y,z>0$ (no condition imposed) there ...