Alexandrov geometry is the study of the geometry of Alexandrov spaces, which are non smooth analogues of Riemannian manifolds with curvature bounded from below.

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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 ...
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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 ...
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87 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 ...
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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 $$ ...
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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 ...
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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 ...
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1answer
168 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 ...
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301 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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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 ...
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355 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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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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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 ...
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97 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 ...
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1answer
174 views

CAT(K) and Busemann [closed]

Can a Busemman space be CAT(1)?
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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 ...
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198 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 ...
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1answer
92 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.
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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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102 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 ...
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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
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303 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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291 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 ...
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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?
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examples of totally geodesic subset

Could you give examples of totally geodesic subset of codim>1 in positively curved Alexandrov space?
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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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332 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 ...
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1answer
196 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 ...
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268 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 ...
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263 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?
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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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258 views

Connecting Lemma in the Alexandrov's existence theorem.

At the moment I am polishing my lecture notes which in particular cover Alexandrov's existence theorem. Denote by $\mathbf{P}_k$ the space of isometry classes of polyhedral metrics on the $\mathbb ...
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Alexandrov geometry techniques for Finsler manifolds.

Hi, first I would like to apologize for my English. It's not my native language and I feel my grasp of it is limited. I've been reading Burago's book on metric geometry and I've that it mentions ...
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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 ...
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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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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 ...
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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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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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Is $SL(n,\mathbb{Z})$ a CAT(0) group?

Is it possible to find a CAT(0) space on which the matrix group $SL(n,\mathbb{Z})$ acts properly discontinuously and cocompactly? Note: when the cocompactness is dropped , it is possible.
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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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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-Lipschitz) + (length-preserving) = isometry

I am looking for an elementary way to prove the following theorem. Theorem. Let $\alpha$ and $\beta$ be two simple convex closed curves in $\mathbb R^2$. Assume $$\mathop{\rm length} ...
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1answer
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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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Minimizing geodesic on a convex surface

Let $\Sigma$ be a smooth convex surface in Euclidean 3-space and $\gamma$ be a unit speed minimizing geodesic in $\Sigma$. Assume that for some $a < b < c$, we have ...
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When is a extension of $\mathbb{Z}$ by a free group a CAT(0) group?

The question has an easy answer, if one replaces free by free abelian: Then the resulting group is always solvable and a solvable subgroup of a CAT(0) group is virtually abelian. If the resulting was ...
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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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Braid groups acting on CAT(0)-complexes

Does the braid group $B_n, n\ge 3$, act properly by isometries on a CAT(0) cube complex? Update 1. During a recent talk of Nigel Higson in Pennstate Dmitri Burago asked whether the braid groups are ...
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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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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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Contracting a geodesic on a space of curvature less than 1

I would like to ask for a reference to the following statement (hopefully correct): Let $M$ be a manifold of sectional curvature at most $1$ and let $\gamma$ be a closed geodesic. Suppose that ...
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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 ...