# Tagged Questions

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

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### Hausdorff convergence of submanifolds in Riemannian manifolds

Let $(M^n,g)$ be a smooth compact Riemannian manifold. It is well known that any sequence $\{X_i\}$ of compact subsets of $M$ has a subsequence which converges in the Hausdorff metric to a compact ...
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### Joint point of coarse geometry and dynamical system?

My major interest is on dynamical systems, but I did REU in a coarse embedding problem. I wonder whether there's some significant connection between those two subjects. I've tried to google for a ...
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### What is the maximal convex hull in $\mathbb R^3$ of a tree with fixed total length?

Denote by $\mathcal T_n$ the set of all trees on $n$ nodes. For a tree $T\in\mathcal T_n$, we assign to each edge a non-negative length such that the sum of all lengths is 1. Denote by $v(T)$ the ...
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### Volume growth of balls

Let $G$ be a locally compact group and $K\subset G$ a compact subgroup. Suppose that on the homogeneous space $X=G/K$ we have a $G$-invariant proper metric $d$. For $R>0$ let $B(R)$ be the open ...
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### Lamination as limit of arcs

I am reading Bonahon's notes on closed curves, in particular the part about hyperbolic laminations. In his notes Bonahon illustrates some examples as why laminations should be "limit curves" on ...
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### Classification of 2-dimensional Alexandrov spaces

Is it possible to classify explicitly compact 2-dimensional Alexandrov spaces with curvature bounded below (either with or without boundary)? If yes, a reference would be helpful. EDIT: If the ...
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### Which surfaces admit unbounded-length simple geodesics?

Let $S$ be a surface embedded in $\mathbb{R}^3$. A simple geodesic on $S$ is one that does not self-intersect. Some surfaces have simple geodesics whose length exceeds any given bound $L$. For example,...
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In notes here http://math.univ-lyon1.fr/homes-www/gille/prenotes/lens.pdf on page $2$ a formulation of distance between two positive quadratic form $[q],[q']$ is given by $$d([q],[q'])=\frac{\sup_{x\... 1answer 116 views ### Geometry of convex subsets in Alexandrov space/ Riemannian manifold Let X^n be an n-dimensional complete Alexandrov space with curvature bounded below (or a smooth Riemannian manifold, possibly with boundary). Let U\subset X be an open dense subset with the ... 1answer 137 views ### Are the primary parallelotopes classified? (equivalently, Voronoi cells of lattices) A primary parallelohedron is a polyhedron that can fill space with infinite translated copies. It is known (e.g., Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 29-30, 1973; or, ... 1answer 68 views ### Optimal covering with finite subcollection of open sets This is mainly a reference request. Consider a finite collection of (let's say, for simplicity) of open balls B_i, i = 1, 2, ..., m in (again, for simplicity) \mathbb{R}^n. I am looking for ... 2answers 179 views ### Volume of the convex hull of the set of all graphic sequences of a given length Consider the set of all graphic sequences with n elements as a subset of \mathbb{R}^{n}, namely let$$D(n)=\{(d_{1},\dots,d_{n})\in\mathbb{Z}_{+}^{n}:d_{1}\geq\dots\geq d_{n},\ \sum_{i=1}^{n}d_{i}\...
Consider a sequence of $n-$dim Alexandrov spaces with curvature $\geq$ -1 $\{(M_i,p_i)\}$ Gromov-Hausdroff converging to an $n-$dim Alexandrov space $(M,p)$. Let $f:M\mapsto \mathbb R$ be a Lipschitz ...
EDIT: Let $X$ be an $n$-dimensional Alexandrov space with curvature bounded below. Let $f_1,\dots, f_n\colon X\to \mathbb{R}$ be $\lambda$-concave functions. Assume that at a fixed point $p$ there ...