# Questions tagged [mg.metric-geometry]

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

2,339 questions
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Sum of squared nearest-neighbor distances between points in a square

Let $\square_2=\{(x,y): 0\leq x, y\leq1\}$ be the unit square in $\mathbb{R}^2$. Take $n>1$ points $P_1, \dots, P_n\in\square_2$. Denote the distances $d_j=\min\{\Vert P_k-P_j\Vert: k\neq j\}$, ...
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### Geodesics on the sphere

In a few days I will be giving a talk to (smart) high-school students on a topic which includes a brief overview on the notions of curvature and of gedesic lines. As an example, I will discuss flight ...
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### The typical amount of lattice points in a set as dimension goes off to infinity

I have encountered a geometry-of-numbers-type problem when calculating an entropy in a lattice communications scheme: $L^{(n)}$ is a lattice uniform over all those in $\mathbb{R}^n$ with base ...
In his seminal paper on hyperbolic groups (see Section 8.1) Gromov defines an isometry $f$ of a hyperbolic space $X$ to be parabolic if the orbit of any point $x\in X$ under the action of \$\langle f\...