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Results tagged with mg.metric-geometry
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user 102946
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
8
votes
Accepted
An extremal property of points on the unit sphere of a 2-dimensional Banach space
The answer is no, in general.
For a counterexample, consider the $\ell^p$-norm on $\mathbb{R}^2$ with $p=4$, and let $x = e_1 = (1,0)$.
We first note that the vectors $e_2 = (0,1)$ and $-e_2$ do not …
- 12.5k
26
votes
Accepted
What are the matrices preserving the $\ell^1$-norm?
As pointed out by YCor in the comments, the following theorem is true:
Theorem 1 Let $p \in [1,\infty] \setminus \{2\}$. If a matrix $A \in \mathbb{R}^{n \times n}$ is an isometry on $\mathbb{R}^n$ w …
- 12.5k