# Questions tagged [mg.metric-geometry]

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

2,341 questions
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### Adjoint of the Hodge de Rham star operator under the integral pairing

Given a Riemannian manifold $(M, g)$ of dimension $n$, the Hodge star operator $\star: \Omega^k(M) \to \Omega^{n-k}(M)$ is defined. What is the (formal) adjoint of $\star$ under the integration ...
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### Integrating over a hypercube, not a hypersphere

Denote $\square_m=\{\pmb{x}=(x_1,\dots,x_m)\in\mathbb{R}^m: 0\leq x_i\leq1,\,\,\forall i\}$ be an $m$-dimensional cube. It is all too familiar that $\int_{\square_1}\frac{dx}{1+x^2}=\frac{\pi}4$. ...
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### Does $G$ act 2-transitively on its Bruhat-Tits building?

Let $k$ be a finite extension of $\mathbb{Q}_p$ and let $G$ be a semisimple Lie group over $k$. We consider the action of $G$ on its Bruhat-Tits building $X$. Question: If $x,y,x',y'$ are vertices, ...
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### 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 ...
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### 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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### An new equilateral triangle related to the Morley triangle

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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### 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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### Why the VC dimension of triangles in 2D space is not greater than 7?

I understand that there are sets of 7 points on a circle that can be fully shattered using triangles.But, it is not clear to me why it cannot shatter 8 points. Is there any intuitive way of arriving ...
I consider convex polytopes $P\subseteq\Bbb R^d$ (convex hull of finitely many points) which are arc-transitive, i.e. where the automorphism group acts transitively on the 1-flags (incident vertex-...