# Questions tagged [metric-geometry]

The study of geometry of manifolds without appealing to differential calculus. It includes studies of length spaces, Alexandrov spaces, and CAT(k) spaces. The techniques are often applicable to Riemannian/Finsler geometry (where differential calculus is used) and geometric group theory. For questions about plain-old metric spaces, please use (metric-spaces) instead.

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### Is every coarse map between proper geodesic spaces a quasi-isometric embedding?

Let $(S,d_{1})$ and $(S',d_{2})$ be two proper geodesic metric spaces. Is every coarse map $f: S\rightarrow S'$ a quasi-isometric embedding?. Just to recall, a coarse map $f: S\rightarrow S'$ between ...
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### K-convexity, duality and in Banach Spaces

I am interested in K-convexity as defined in https://webusers.imj-prg.fr/~bernard.maurey/articles/typandco.pdf Page 24 bottom of page. I am struggling to show that K-convexity passes to the dual given ...
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### Geodesic convexity of small balls in Alexandrov CBB spaces

If $X$ is a Riemannian manifold, it is known that, for $p\in X$, there is some $\epsilon>0$ such that $B(p,\epsilon)$ is geodesically convex. Geodesically convex means there is a minimizing ...
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### When does the union of all geodesics equal the metric interval?

Definitions: Throughout let $(M,d)$ be a geodesic metric space [cf. p. 104 of EoD]* with $d$ the (strictly) intrinsic metric, i.e. for all $x,y \in M$, $d(x,y)$ equals the length of any minimizing ...
49 views

### Spectral radius of a finitely generated group

Let $G$ be a finitely generated group and $\Gamma$ be its Cayley graph with the usual word metric. Let $\mu$ be a non-degenerate measure on $G$, and define $p(x, y) = \mu(x^{-1} y)$. As is well-known, ...
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### Proof that the sum of the exinradius is equal to 4 times the circunradius and the exinradius.

I have seen a problem that uses this identity in his solution however I haven't seen anywhere a proof of this lemma. I have read that is called steiner relation. I think that maybe carnot's theorem ...
39 views

### Algebraic Characterisation of the End Space of a proper geodesic space in terms of non-continuous functions

Let $X$ be a rimcompact Tychonoff space, that is, a completely regular Hausdorff space with a base of open sets with compact boundaries. It is well known that $X$ has a maximal compactification with a ...
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### What is the degeneracy of the "Coxeter plane" for the E8 lattice?

The 240 minimal vectors (roots) of the E8 lattice, projected onto "the" Coxeter plane, are shown here: https://en.wikipedia.org/wiki/E8_(mathematics)#/media/File:E8Petrie.svg and discussed ...
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### Examples of CAT(0)-spaces that are no manifolds

I´m trying to think of some examples of CAT(0)-spaces that are not manifolds. I haven´t found any examples in the book of Bridson and Haefliger, so I have tried to come up with some own examples. I ...
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