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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
        57 views

        Asymptotic bound on minimum epsilon cover of arbitrary manifolds

        Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\varepsilon)$ denote the minimal cardinal of an $\varepsilon$-cover $P$ of $M$; ...
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        votes
        1answer
        70 views

        Reference Request: Carnot Group Not Containing Group of Isometries

        This question is a follow-up to this post, from which I quote: Let $\mathfrak{e}$ be the 3-dimensional Lie algebra with basis $(H,X,Y)$ and bracket $[H,X]=Y$, $[H,Y]=-X$, $[X,Y]=0$. It is ...
        1
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        0answers
        36 views

        Describing hull of vertex intersections of two convex bounded polytopes?

        We have two convex bounded polytopes $P_1$ and $P_2$ where a. $P_2\subseteq P_1$ b. $\mathcal{V}(P_2)\cap\mathcal{V}(P_1)\neq\emptyset$. Is there a name for the polytope $P=\mbox{Conv}(\mathcal{V}(...
        5
        votes
        0answers
        109 views

        How to calculate the volume of a parallelepiped in a normed space?

        Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a ...
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        votes
        0answers
        16 views

        Non-collapsed Alexandrov spaces, level surface of regular map homeo to its lifting?

        Let $X_i$ be n dimensional, no boundary Alexandrov spaces with curvature $\geqslant -1$ and diameter $\leqslant D$. Suppose that $X_i$ converge to an n dimensional Alexandrov space $X$. Then by ...
        5
        votes
        0answers
        136 views

        Area of $n$-sphere contained outside $\ell_1$ ball

        For a given $r>1$, what is the surface area of $\mathbb S^{n-1}$ (the sphere of radius 1 in $\mathbb R^n$) which is contained outside of the $\ell_1$ ball of radius $r$? Or equivalently, if $X\sim ...
        4
        votes
        0answers
        70 views

        Discrepancy of the finite approximation of the Lebesgue measure

        Let $\mu$ be a probabilistic measure on the unit square $Q$ which is the average of $N$ delta-measures in some points in this square; let $\lambda$ denote the Lebesgue measure on $Q$. What is the rate ...
        7
        votes
        2answers
        471 views

        If a (distance) metric on a connected Riemannian manifold locally agrees with the Riemannian metric, is it equal to the induced metric?

        Let $(M,g)$ be a connected Riemannian manifold. Let $d_g$ be the induced distance metric of $g$. Now let $d$ be some other metric on $M$. Suppose that for each $x \in M$, there is a neighborhood $U$ ...
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        vote
        0answers
        38 views

        Existence of $1$-Lipschitz map between triangles

        Crosspost from math.SE Consider two (Euclidean) triangles $T$ and $T'$. Let's say that $T$ majorizes $T'$ if there exists a 1-Lipschitz map that sends vertices to vertices and sides to sides (for ...
        5
        votes
        3answers
        436 views

        Alexandrov's generalization of Cauchy's rigidity theorem

        Wikipedia states that A. D. Alexandrov generalized Cauchy's rigidity theorem for polyhedra to higher dimensions. The relevant statement in the article is not linked to any source. The sources at the ...
        3
        votes
        0answers
        88 views

        Hyperbolic Intercept (Thales) Theorem

        Is there an Intercept theorem (from Thales, but don't mistake it with the Thales theorem in a circle) in hyperbolic geometry? Euclidean Intercept Theorem: Let S,A,B,C,D be 5 points, such that SA, SC, ...
        8
        votes
        1answer
        475 views

        Illustrating that universal optimality is stronger than sphere packing

        I'm a physicist interested in the conformal bootstrap, one version of which was recently shown to have many similarities to the problem of sphere packing. Sphere packing in $\mathbf{R}^d$ has been ...
        1
        vote
        1answer
        118 views

        Does fractallity depend on the Riemannian metric?

        Edit: According to comment of Andre Henriques we revise the question: In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that ...
        2
        votes
        1answer
        67 views

        The space of complex structure compatible with metric

        Why the space of all complex structure on a $2n$-dimensional vector space which compatible with a positive definite metric is diffeomorphic to $ O(2n)/U(n) $ ?
        10
        votes
        1answer
        290 views

        Growing a chain of unit-area triangles: Fills the plane?

        Define a process to start with a unit-area equilateral triangle, and at each step glue on another unit-area triangle.                     $50$ ...

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