# 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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### Must a Geodesic Metric Space be a Length Space?

A metric space $(X,d)$ is said to be a geodesic metric space if and only if each pair of points $x,y \in X$ is connected by a geodesic, where the geodesic need not be unique. I assume that having this ...
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### Almost-isometric embedding is almost surjective

I'm looking for a statement, if it exists, of the following sort: Let $X$ be a compact metric space. Let $f:X\to X$ be a function, and suppose that for some (fixed) $\epsilon>0$, for all $x,y\in X$...
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### Using the Rodrigues' rotation formula to rotate a parametric surface?

I am currently trying to solve a problem that involves rotating a parametric surface in 3D space. The surface is a Cone, with parameterisation $C=r \cos(a)i + r \sin(a)j + rk$. To do this I decided to ...
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### Convergence of Subsequences in Gromov's Theorem

A key theorem in metric geometry is Gromov's Compactness Theorem, which describes when a sequence of metric spaces has a Gromov-Hausdorff converging subsequence. I'm not sure I follow how this works: ...
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### What is a plane figure covering all sets of diameter $1$?

Problem : (1) Show that there is a plane figure $F$ of least area which is capable of covering any plane figure of unit diameter. (2) Try to guess what is $F$. Proof of (1) : Define $\mathcal{H}$ to ...