# Tagged Questions

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

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### The average length of vectors on spherical caps

Let $U$ denote the uniform distribution on the $n$-ball of radius $1$. What is the expected square-length of a vector under this distribution: $$\mathbf{E}_U[\|x\|^2]$$ By standard concentration-...
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### Three homothetic centers are collinear

I am looking a proof for the problem as follows: Let a hexagon, such that its principal diagonals are concurrent. For each side of the hexagon, extend the adjacent sides to their intersection, ...
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### Uniform sampling from general simplex with a twist

This is part of a question I had asked elsewhere, and then some of the links redirected me to CS stack exchange. Given $0\leq a_1\leq\dots\leq a_D\leq1$ (all strictly positive), I want to draw points ...
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### asymptotic behavior of Lipschitz constants of sectional curvature

I'm studying the paper "Measure Concentration and the Topology of Positively-Curved Riemannian Manifolds" (https://arxiv.org/pdf/1402.4947v1.pdf) and I have some problem in understanding the proof of ...
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### Mathematical Definition of $n$-Brouillin Zone [duplicate]

I am having trouble finding a mathematical definition of the Brouillin zone beyond the first, which are basically the Voronoi cells or Wigner-Seitz cells. We could imagine the set of point closer to ...
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### Negative-curvature behaviour of higher-rank symmetric spaces

Let $X$ be a symmetric space of noncompact type with $rk\ X\geq 2$ and let $G$ be the identity component of the isometry group. Pick points $p\in X$ and $\xi\in\partial_{\infty}X$, with $\xi$ regular, ...
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### Area of a polyellipse

Pretty much like the title asks. I'll explain the situation. I have a set of points $N \in \mathbb{R^2}$. To select a next point $n_i$ to add to the set, select the node with the smallest combined ...
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### CAT(0)-groups in dimension 2

Suppose I have a space $X$ which is connected, simply connected, CAT(0) of dimension 2 and a group $G$ which acts on $X$ freely, isometrically, properly discontinuously and cocompactly. What can be ...
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