**2**

votes

**1**answer

68 views

### Set of regular points in an Alexandrov space with curvature bounded below

Let $X^n$ be an $n$-dimensional Alexandrov space with curvature bounded below. A point $x\in X$ is called regular if the space of directions $\Sigma_x$ is isometric to the standard sphere $S^{n-1}$.
...

**1**

vote

**0**answers

40 views

### Doubling theorem for Alexandrov spaces

Is there a user friendly exposition of the notion of boundary of an Alexandrov space with curvature bounded from below and of the Doubling theorem?
The only reference I am aware of is the original ...

**1**

vote

**0**answers

53 views

### A compact Alexandrov space with curvature bounded below has curvature bouneded above?

For a compact Riemannian manifold, Since the curvature tensor is continuous, we know that the sectional curvature is bounded, i.e. bounded above and below. Now let $M$ be a compact Alexandrov space ...

**4**

votes

**0**answers

85 views

### Gromov's compactness theorem for manifolds with boundary

The Gromov's compactness theorem says that if $\{M_i^n\}$ is a sequence of closed Riemannian manifolds of dimension $n$ with uniformly bounded diameter and uniformly bounded from below Ricci curvature ...

**0**

votes

**0**answers

22 views

### Concave functions on a cone over Alexandrov space

Let $\Sigma$ be an Alexandrov space of curvature $\geq 1$ without boundary. Let $X$ be the cone over $\Sigma$. $X$ is well known to be non-negatively curved.
Let $f\colon X\to \mathbb{R}$ be a ...

**1**

vote

**1**answer

53 views

### Two geodesics with angle $\pi$ in Alexandrov space

Let two geodesic segments in an Alexandrov space with curvature bounded from below start at the same point and the angle between them equals $\pi$. It is possible that these segments are not the two ...

**1**

vote

**1**answer

88 views

### Gradient of distance function at cut points on Alexandrov spaces

Let $M$ be an $n$-dim Alexandrov space with curvature bounded below $sec \geqslant k$, possibly non-compact. We assume that $M$ has no boundary for simplicity. For a compact subset $K \subset M$, the ...

**3**

votes

**1**answer

81 views

### Gauss-Bonnet formula for 2-dimensional Alexandrov spaces

EDIT: Let $S$ be a closed orientable 2-dimensional surface equipped with a metric with curvature $\geq \kappa$ in the sense of Alexandrov.
Questions 1. Can one define a measure $K$ on $S$ (thought ...

**6**

votes

**2**answers

122 views

### Geodesics on convex hypersufaces

Let $M^n$ be the boundary of a convex compact set in $\mathbb{R}^{n+1}$ with non-empty interior.
Question 1. Is $M$ geodesically complete, i.e. is it true that every geodesic (= locally shortest ...

**6**

votes

**2**answers

132 views

### Alexandrov spaces which are not limits of Riemannian manifolds

Are there important/ interesting/ natural examples of compact Alexandrov spaces with curvature bounded from below which are not Gromov-Hausdorff limits of smooth compact Riemannian manifolds with ...

**3**

votes

**0**answers

41 views

### Convex hulls of quasi-convex sets in proper CAT(0) spaces

Let $A$ be a quasi-convex set in some proper CAT(0) space, $X$, and let $\mbox{Hull}(A)$ be the intersection of all convex sets containing A. Can we conclude that $\mbox{Hull}(A)$ is in some bounded ...

**8**

votes

**0**answers

87 views

### What are the extremal CAT(0) metrics?

(Split off from Does every CAT(0) space embed in a product of trees? )
Fix an integer $k \ge 2$, and let
$MC0_k \subset \mathbb{R}^{\binom{k}{2}}$ be the set of possible squared-distances between $k$ ...

**11**

votes

**2**answers

297 views

### Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees?

Question 1. Does every CAT(0) space embed isometrically inside an integral of $\mathbb{R}$-trees?
Here an integral of $\mathbb{R}$ trees means the set of functions from a measure space $\mathcal{F}$ ...

**1**

vote

**1**answer

75 views

### Alexandrov spaces of constant curvature

Let $X$ be locally compact Alexandrov space whose curvature satisfies both inequalities $\geq K$ and $\leq K$. What can be said about such a space? Is it locally isometric to the standard Riemannian ...

**0**

votes

**1**answer

70 views

### Existence of shortest paths in complete Alexandrov spaces

Let $X$ be complete finite dimensional Alexandrov space with curvature bounded from below. Is it true that any two points can be connected by a shortest path? If this is not true in general, it it ...

**2**

votes

**1**answer

121 views

### Shortest paths in Alexandrov spaces

Let $X$ be an Alexandrov space with curvature bounded from below (if necessary, $X$ might be assumed to be finite dimensional or even compact).
Question 1. Is it true that every point of $X$ has a ...

**3**

votes

**1**answer

156 views

### Codimension of the set of topologically singular points of an Alexandrov space.

I am reading Burago, Burago and Ivanov's book A course in metric geometry. In chapter 10 the mention that Alexandrov spaces of curvature bounded below have a stratification into topological manifolds. ...

**2**

votes

**1**answer

102 views

### Why finite dimensional MCS-space implies $\mathbb{R}^m\times cone$ locally?

Frequently, I see a statment of the result in reference that a finite dimensional Alexandrov spaces is locally a product $\mathbb{R}^m\times cone$. It seems that this result comes from Perelman's ...

**3**

votes

**1**answer

75 views

### Whether the manifold part of an Alexandrov space is connected?

The title is my question.
Alexandrov space here means finite dimensional Alexandrov space with curvature bounded below ,denoted by CBB.
Let $\gamma$ be a simple curve in a $n$ dimensional CBB $M$ ...

**1**

vote

**0**answers

112 views

### A question about a manifold in an $n$-dimensional Alexandrov space with curvature bounded below [duplicate]

Suppose $M$ is an $n$-dimensional Alexandrov space with curvature bounded below(maybe with boundary), subspace $A\subset M$ is an $n$-dimensional manifold without boundary. Then whether every point in ...

**3**

votes

**0**answers

88 views

### Surfaces with curvature $\leq K$ are of bounded integral curvature

One characteristic of a CBA($K$) surface (a topological surface with an intrinsic metric of curvature $\leq K$ in the sense of Alexandrov) is that $\delta_K(T) \leq 0$, where $\delta_K(T)$
is the ...

**5**

votes

**2**answers

238 views

### Are shortest halving curves simple closed geodesics?

Let $S$ be a smooth convex surface in $\mathbb{R}^3$
(although my question may as well be asked for the surface of a polyhedron).
Say that $\gamma$ is a shortest halving curve if
(a) it partitions the ...

**2**

votes

**0**answers

98 views

### Cusp points in Alexandrov spaces

Given a space of bounded integral curvature (by which I mean a topological surface with an intrinsic metric, such that the sum of excesses of any finite collection of non-overlapping simple triangles ...

**2**

votes

**0**answers

79 views

### isoperimetric problems on Alexandrov spaces

For an Alexandrov space M with curvature bounded from below, the isoperimetric profile $v \to I_M(v)$ defined for every $v\in (0,V(M))$ (the volume of M might be infinite), is given by
$$
...

**0**

votes

**1**answer

66 views

### The set of strained points in an Alexandrov space is open [closed]

I'm reading Burago, Burago and Ivanov's book, and I'm on the section about Strainers. The authors say that it is obvious that the set of $(m,\varepsilon)$-strained points for any fixed natural number ...

**2**

votes

**0**answers

151 views

### Convex functions with non-singular hessian measure are continuously differentiable?

It is known that every convex function $f: \Omega\to \mathbb{R}$, $\Omega$ convex subset of $\mathbb{R}^n$, has a weak derivative of bounded variation $Df\in BV_{loc}(\mathbb{R}^n)$ (e.g. Evans and ...

**7**

votes

**1**answer

276 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 ...

**11**

votes

**1**answer

345 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 - ...

**5**

votes

**0**answers

115 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 ...

**14**

votes

**3**answers

408 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 ...

**1**

vote

**0**answers

86 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 ...

**1**

vote

**0**answers

66 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**

votes

**0**answers

106 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 ...

**-1**

votes

**1**answer

207 views

**0**

votes

**0**answers

85 views

### Connected sum in Alexandrov spaces

Is it possible to take connected sums of Alexandrov spaces? More explicitly, can one put a metric that turns the connected sum into an Alexandrov space? Does it matter if the curvature bound is from ...

**8**

votes

**2**answers

220 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 ...

**1**

vote

**1**answer

98 views

### examples of space of direction at a point in an infinite dim Alexandrov space compact

The space of direction at a point in an infinite dim Alexandrov space can be compact?Please give examples or prove it's wrong.

**4**

votes

**1**answer

178 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 ...

**0**

votes

**0**answers

106 views

### Must a hyperbolic cone over Riemannian manifold be manifold?

M is a hyperbolic cone over an n-1 dim Riemannian manifold N with $Ric(N) \ge - \left( {n - 2} \right)$ ie
$M = R \times {}_{\cosh \left( t \right)}N$,Surely N is an Alexandrov space,must M be a ...

**0**

votes

**0**answers

69 views

### examples of Alexandrov space with sec>=-1 and first eigenvalue =(n-1)^2/4

could someone give some examples :nonRiemannian manifold Alexandrov space with sec>=-1 and the first eigenvalue equal to (n-1)^2/4

**7**

votes

**1**answer

314 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 ...

**0**

votes

**1**answer

296 views

### about parabolic cone

I want to prove some Alexandrov space M is parabolic cone X x R.Since Alex has no Riemannian metric,so how to do?Is there any (triangle) formula about the relation of distance of two points in M and ...

**0**

votes

**0**answers

101 views

### Is level set of Busemann function on Alexandrov space again Alexandrov space?

M is an Alexandrov space with curv>=-1,containing a line(ray).Is level set of Busemann function on M again Alexandrov space?If not,can you give a counterexample?

**-2**

votes

**2**answers

234 views

### examples of totally geodesic subset

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

**1**

vote

**1**answer

526 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 ...

**13**

votes

**1**answer

356 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 ...

**0**

votes

**1**answer

209 views

### positively curved Alexandrov space

I heard a conjecture "3-dim positively curved Alexandrov space is of the form S^3/J.(I cannot make sure my statement is accurate).
What is the classification of n-dim positively curved Alexandrov ...

**4**

votes

**1**answer

280 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 ...

**5**

votes

**1**answer

286 views

### 3-dim positively curved Alexandrov space

What is the classification of 3-dim positively curved Alexandrov space?
And if a 3-dim positively curved Alexandrov space has a totally (quasi)geodesic subset,then the classification?

**7**

votes

**2**answers

378 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 ...