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

        4
        votes
        2answers
        88 views

        Function as sum of distances over a connected, compact metric space

        If $X$ is a connected, compact metric space with distance function $d : X^2 \rightarrow \mathbb{R}^+$, is it true that there exists a positive real number $a$, dependent on $X$ and $d$, such that for ...
        16
        votes
        1answer
        243 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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        votes
        0answers
        54 views

        Equilateral triangle related to Morley triangle. Which is the barycentric coordinate of the perspector?

        Morley equilateral triangle is the nice theorem in Eulidean Geometry. I found an equilateral triangle and a group circle related to the Morley triangle and angle trisectors: Let $ABC$ be a triangle ...
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        vote
        0answers
        59 views

        Some hypersurface has a positive second fundamental form potentially

        Notation : $r^2=x^2+y^2$. Exercise : Define $$F_\sigma (x,y)= (f_\sigma (x,y),\frac{x^2-y^2}{2},xy)$$ where $$f_\sigma (x,y)=(1- \sigma^2 r^2)(x-\sigma x^3,y-\sigma y^3)$$ Define $ G_\sigma: \mathbb{...
        5
        votes
        0answers
        86 views

        Do manifolds with non-negative Ricci curvature allow bi-Lipschitz embeddings into Euclidean spaces?

        QUESTION: Let $n$ be a natural number. Is it true that there exist $N(n), D(n) > 0$ such that any complete $n$-dimensional Riemannian manifold of nonnegative Ricci curvature can be embedded into $N$...
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        votes
        2answers
        135 views

        largest diameter of intersection of two balls

        Two closed balls with a common radius are positioned so that the centre of either ball is on the boundary of the other. I am interested in the extremal diameter of their intersection, in an arbitrary ...
        4
        votes
        0answers
        112 views

        Non-algebraic quasi-isometric embeddings

        What are examples of finitely generated groups $\Gamma$ and $\Lambda$ such that the metric space $\Lambda$ embeds into $\Gamma$ quasi-isometrically but such that $\Lambda$ is very much not a subgroup ...
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        votes
        0answers
        168 views

        Are these points known? [closed]

        Let $ABC$ be a triangle and $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $A'$, $B'$, $C'$ respectively. From my construction by GeoGebra, I found two special points as ...
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        vote
        0answers
        70 views

        DIstance on a Riemannian manifold [closed]

        Given a Riemannian manifold $(M,g)$ is it possible to calculate the distance between two points on this manifold. Is it possible the inverse? That means: given a formula of the distance, for example: ...
        6
        votes
        0answers
        84 views

        Geometric mean of three or more positive definite matrices

        The geometric mean of two positive definite (Hermitian) matrices of same size is defined by $$A\natural B := A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$$equivalently, $$A\natural B =(BA^{-1})^{1/2}A=A(A^...
        4
        votes
        0answers
        136 views

        Uniqueness of the boundary of a hierarchically hyperbolic group

        Hierarchically hyperbolic groups and spaces (HHG and HHS for short) were defined by Behrstock, Hagen and Sisto (see here and here). Examples include mapping class groups, Right angled Artin groups, ...
        11
        votes
        0answers
        198 views

        Why are the medians of a triangle concurrent? In absolute geometry

        This fact holds true in absolute geometry, and I would like to see an elementary synthetic proof not using the classification of absolute planes (Euclidean and hyperbolic planes) and specific models. ...
        0
        votes
        0answers
        54 views

        Which $CAT(0)$-polygonal complexes are median spaces?

        $CAT(0)$-polygonal complexes are simply connected collections of glued (on their faces) polyhedra of varying dimension such that each link is flag. Which $CAT(0)$-polygonal complexes with appropriate ...
        3
        votes
        1answer
        123 views

        Are $CAT(0)$-polygonal complexes median spaces?

        A median space is a metric space $X$ for which for any three points $x, y , z \in X $ there exists a unique point $m$ such that $d(x,m)+ d(m, y)= d(x , y ), d(x,m)+ d(m, z)= d(x , z ), d(y,m)+ d(m, z)=...
        1
        vote
        1answer
        126 views

        Approximate the following series on the euclidean grid

        I had trouble in finding a closed form solution for the following series, so now I am trying to find a good approximation for it. The $\sqrt{i^2 + j^2}$ in the exponent comes from distances on the ...

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        山西福彩快乐十分钟
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