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      Questions tagged [mg.metric-geometry]

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

      2
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      0answers
      34 views

      Asymptotically discrete ultrafilters

      Definition 1. A ultrafilter $\mathcal U$ on $\omega$ is called discrete (resp. nowhere dense) if for any injective map $f:\omega\to \mathbb R$ there is a set $U\in\mathcal U$ whose image $f(U)$ is a ...
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      12 views

      The stability of optimal transport ground cost matrices

      I would like to better understand the stability of the ground cost matrix $C \in \mathbb{R}^{n \times n}_+$ within the discrete Kantorovich optimal transport problem: $$\mathcal{P}(C) := \arg \min_{P ...
      3
      votes
      1answer
      175 views
      +50

      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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      votes
      1answer
      34 views

      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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      votes
      0answers
      168 views

      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 ...
      3
      votes
      0answers
      44 views

      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 ...
      2
      votes
      0answers
      57 views

      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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      votes
      2answers
      270 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 ...
      5
      votes
      0answers
      99 views

      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, ...
      1
      vote
      1answer
      97 views

      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 ...
      2
      votes
      1answer
      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 ...
      1
      vote
      1answer
      75 views

      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 ...
      2
      votes
      1answer
      124 views

      $L^{2}$ 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 ...
      5
      votes
      0answers
      96 views

      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 ...
      2
      votes
      1answer
      78 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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