# Tagged Questions

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**1**answer

33 views

### Length inequalities in trees and CAT(0) spaces

I have a family of possibly related questions. Let me start with an elementary one:
Question 1. Fix an integer $n$. For which collections of real numbers $a_{ij}$, $i, j = 1, \dots, n$, is it true ...

**10**

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**1**answer

255 views

### Tverberg's theorem in CAT(0) spaces

Does Tverberg's theorem hold for CAT(0) spaces of covering dimension $d<\infty$:
Is it true that for any $d$-dimensional $CAT(0)$-space $X$ and a subset $E\subset X$ of cardinality $(d + 1)(r - ...

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**0**answers

99 views

### Fourier analysis for the discrete cube in CAT(0) spaces?

Is there a meaningful Fourier analysis of mappings from the discrete cube into CAT(0) spaces?
Examples for what I have in mind:
Fix a CAT(0) space $X$, a mapping $f:\{-1,1\}^n \to X$, and ...

**12**

votes

**3**answers

327 views

### Embedding expanders in CAT(0) spaces

It is well-known that expanders are hard to embed into Hilbert (or $\ell^p$) spaces - any embedding of an expander with $n$ vertices has distortion $\Omega(\log n)$.
Can anyone provide a reference ...

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**0**answers

75 views

### Alexandrov spaces satisty $BE(K,N)$ and $BE(K,\infty)$?

Assume the Dirichlet form $\varepsilon$ adimits a Carre du champ $\Gamma$ and introduce the multilinear form $\Gamma_2$ $$
\Gamma_2 [f,g;\phi]:=\frac12 \int_X (\Gamma (f,g)L\phi ...

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**0**answers

45 views

### curvature of subset of Alexandrov spaces

If M is a Riemannian manifold with $Ric \ge - \left( {n - 1} \right)$, $$ds_M^2 = d{t^2} + \exp \left( {2t} \right)ds_N^2$$ N is a submanifold of M. Then by Gauss-equation, we can prove $Ric\left( N ...

**3**

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**0**answers

91 views

### Generalized flag complex?

Assume we glue an $n$-dimensional simplicial complex $K$
from copies of an $n$-simplex $\Delta$ with fixed spherical metric.
We may think that $\Delta$ has colored vertices
and we glue so that the ...

**7**

votes

**2**answers

196 views

### Distortion of tree embedding in Alexandrov spaces

It is a well-known theorem first proved by Bourgain that any map $\varphi:T_n\to H$ from the binary tree of height $n$ to a Hilbert space has distortion at least $C \sqrt{\ln n}$ where $C$ is a ...

**0**

votes

**0**answers

38 views

### curvature of cone over a cuboid bounded below?

As we know,if M is an Alexandrov space with sec>=1,then the cone over M has sec>=0.What if when M is a cuboid with side length r1,...,rn,dia(M)<=π,then the cuvature of the cone over M bounded below ...

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152 views

### Does convex set in Alexandrov space has positive reach?

Let $M$ be a metric space, $A$ a subset of $M$. The reach (defined by Federer) of $A$ in $M$ is the largest $r_0\ge 0$ such that if $x\in M$ and the $d(x, A)< r_0$, then $A$ contains a unique point ...

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**1**answer

280 views

### Is the tangent cone of a totally convex subset again totally convex?

To need not worry about the possibly broadest context let:
$X$ be an Alexandrov's space with lower curvature bound and $C$ be a totally convex subset, i.e. for any $x,y \in C$ and any geodesic ...

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**2**answers

225 views

### examples of totally geodesic subset

Could you give examples of totally geodesic subset of codim>1 in positively curved Alexandrov space?

**1**

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**1**answer

490 views

### a result of soul theorem,right?

X is an n-dim positively curved manifold and Y is a totally geodesic submanifold of codimension 1.Then cutting along Y we get n-dim positively curved manifolds without boundary,by soul theorem these ...

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**1**answer

322 views

### Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space?

Let $X$ be a complete CAT$(0)$ metric space, and $\partial X$ its boundary.
One way to define $\partial X$ is as the equivalence class of geodesic rays
$\gamma(t), \gamma'(t)$
that remain within a ...

**4**

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**1**answer

257 views

### Extend the Wilking Connectiveity Theorem to Alexandrov spaces

In the conference "on Manifolds with Non-negative Sectional Curvature" held in 2007,
Problem 6 is:
Extend the Wilking Connectivity Theorem to Alexandrov spaces, i.e. if $X$ is a positively curved ...

**6**

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**2**answers

337 views

### Source for: Geodesics in CAT(0) spaces

I am seeking a good introductory reference that could lead to an understanding of
the properties of geodesics in
complete CAT(0) metric spaces.
I am especially interested in learning the differences ...

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**0**answers

254 views

### Is it overkill to invoke Kirszbraun theorem to prove the following fact ?

Given a small enough convex triangle $(abc)$ in a (smooth or Alexandrov) surface $(X,d)$ of curvature greater than $-1$, let $(\overline{abc})$ be its comparison triangle in $\mathbb{H}^2$. Then there ...

**7**

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**1**answer

444 views

### Smoothability of compact Alexandrov surfaces with curvature bounded from below.

Let $(X,d)$ be compact metric space of curvature greater than $-1$ (in the sense of comparison triangles), assume that its Hausdorff dimension is $2$. Then a result of Perelman says that $X$ is a ...

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**1**answer

235 views

### Flat sector in a proper cocompact CAT(0) space

Let $X$ be a complete CAT(0) space with a proper and cocompact group action by isometries, and suppose there are $\xi, \xi' \in \partial X$ with $\angle (\xi, \xi') < \pi$. Using proposition 9.5 ...

**12**

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**2**answers

576 views

### infinite dimensional CAT(0) groups

Usually a CAT(0) group is defined to be a group acting properly isometrically and cocompactly on a CAT(0) space, but I would like to consider only those groups that act properly, isometrically and ...

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**1**answer

294 views

### Examples of CAT(0)-groups

My question is the following:
Let M be a simply connected Riemannian manifold whose sectional curvatures
are all nonpositive and let G be a group. Suppose that G acts in M properly discontinuous and
...

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votes

**3**answers

303 views

### Is this the CAT(0) metric on an affine building?

Let $R$ be a discrete valuation ring qith quotient field $Q$ and let $t\in R$ be a generator of the unique maximal ideal in $R$. Let $V$ be a finite-dimensional $Q$-vector space. Then one can consider ...

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**1**answer

504 views

### Metric spheres in CAT(0) manifolds

Let $X$ be a topological manifold of dimension $n$, equipped with a compatible CAT(0) metric.
Are sufficiently small metric spheres in $X$ homeomorphic to metric spheres in Euclidean space ...

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**1**answer

472 views

### Metrically singular Alexandrov space.

Perelman's stability theorem shows in particular that a finite dimensional compact Alexandrov space $(X,d)$ such that $X$ is not a topological manifold cannot be approximated in the Gromov-Hausdorf ...

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**1**answer

990 views

### Rigidity of triangle comparison in Alexandrov spaces

For $CAT(\kappa)$ spaces $X$ we have following rigidity result: if equality holds in any of the comparison distances between a triangle $\Delta$ in $X$ and the corresponding comparison triangle ...

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**0**answers

532 views

### Are all these groups CAT(0) groups?

Given a geodesic metric space $X$ together with a choice of midpoints
$m:X\times X\rightarrow X$ (i.e. $d(m(x,y),x)=d(m(x,y),y)=d(x,y)/2$).
Assume furthermore, that the following nonpositive ...

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votes

**1**answer

177 views

### Stability of midpoints in CAT(0) spaces

Given a CAT(0) space $X$ and a compact, convex subset $A$ of $X$. One can define its midpoint $m(A)$ as the point, at which the following function attains its minimum.
$f:A\rightarrow ...

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**1**answer

1k views

### Details of Perelman's example about soul of Alexandrov space

Reading Perelman's preprint(1991) Alexandrov space II now. Got confused about the last section 6.4, which contains an example which indicate that the statement ".... manifold is diffeomorphic to the ...

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**1**answer

613 views

### In a locally CAT(k) space, does uniqueness of geodesics imply the lack of conjugate points?

A complete, simply connected Riemannian manifold has no conjugate points if and only if every geodesic is length-minimizing. I just realized that I don't know whether the same is true for a locally ...

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**1**answer

259 views

### Is Level set of Regular functions in Alexandrov spaces again an Alex. space?

Let $X^n$ be an Alexandrov space, and $f: X^n\to \mathbb R^k$ a regular map, does the level set necessary be an Alexandrov space?
In my mind, the intrinsic metric on the level set is 'comparable' to ...

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votes

**2**answers

674 views

### Are isometries the only geodesic preserving maps in a CAT(0)-space?

Given any CAT(0) space $X$, we can define a map $s:X\times X\times [0;1]\rightarrow X$, such that $s(x,y,-)$ is the constant speed geodesic from $x$ to $y$ . Any isometry $f$ of $X$ is compatible with ...

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567 views

### Soul theorem for non-negativly curved open Alexandrov manifolds?

Alexandrov manifold means Alexandrov space which happens to be a manifold, i.e. the space of directions is homeomorphic to shpere. Sorry for introducing this new term.
For such a open manifold does ...

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**2**answers

283 views

### Why is GL(n,C)/U(n) a CAT(0) space?

The title says it all. In one of his answers to the question "Convex hull in CAT(0)" (I don't have the points to post a link, if someone doesn't mind link-ifying this that would be cool), Greg ...

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**2**answers

831 views

### Example of non-closed convex hull in a CAT(0) space

this is related to this question but is simpler, and hopefully is well-known. There are a number of references that say that the convex hull of a collection of points in a CAT(0) space need not be ...