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        Questions tagged [mg.metric-geometry]

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

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        69 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 ...
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        Approximate a Lipschitz function by an affine function [on hold]

        Suppose $\mathcal{F}$ is a Lipschitz function (Lipschitz constant $L$) space defined on $A$. $A$ is a bounded subset of $\mathbb{R^n}$. The $\mathcal{G}$ is an affine function space defined on $A$. ...
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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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        69 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 ...
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        1answer
        120 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 ...
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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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        1answer
        73 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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        169 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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        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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        1answer
        89 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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        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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        1answer
        63 views

        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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        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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        60 views

        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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