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Results tagged with alexandrov-geometry
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user 5010
Alexandrov geometry studies non smooth analogues of Riemannian manifolds with curvature bounded from below or above. It includes spaces with curvature bounded below (briefly $\mathrm{CBB}[\kappa]$) and spaces with curvature bounded above (briefly $\mathrm{CAT}[\kappa]$).
21
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2
answers
683
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Gluing hexagons to get a locally CAT(0) space
I believe that there are four ways to glue (all) the edges of a regular Euclidean hexagon to get a locally CAT(0) space:
The first two give the torus and the Klein bottle, respectively. What are the …
- 10.1k
14
votes
3
answers
750
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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 $\mathcal{F}$ t …
- 10.1k
12
votes
1
answer
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What are the extremal CAT(0) metrics?
(Split off from Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees? )
Fix an integer $k \ge 2$, and let
$MC0_k \subset \mathbb{R}^{\binom{k}{2}}$ be the set of possible squa …
- 10.1k
11
votes
1
answer
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Is the center of gravity in a CAT(0) space contained in the convex hull?
In reading Greg Kuperberg's partial answer to this question Convex hull in CAT(0) ,
I started wondering if the center of gravity is always contained in the closed convex hull.
More precisely, given $ …
- 10.1k
10
votes
1
answer
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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 th …
- 10.1k