235 lines
8.9 KiB
Markdown
235 lines
8.9 KiB
Markdown
# Math 401 Paper 1, Side note 3: Levy's concentration theorem
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## Basic definitions
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### Lipschitz function
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#### $\eta$-Lipschitz function
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Let $(X,\operatorname{dist}_X)$ and $(Y,\operatorname{dist}_Y)$ be two metric spaces. A function $f:X\to Y$ is said to be $\eta$-Lipschitz if there exists a constant $L\in \mathbb{R}$ such that
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$$
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\operatorname{dist}_Y(f(x),f(y))\leq L\operatorname{dist}_X(x,y)
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$$
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for all $x,y\in X$. And $\eta=\|f\|_{\operatorname{Lip}}=\inf_{L\in \mathbb{R}}L$.
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That basically means that the function $f$ should not change the distance between any two pairs of points in $X$ by more than a factor of $L$.
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## Levy's concentration theorem in _High-dimensional probability_ by Roman Vershynin
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### Levy's concentration theorem (Vershynin's version)
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> This theorem is exactly the 5.1.4 on the _High-dimensional probability_ by Roman Vershynin.
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#### Isoperimetric inequality on $\mathbb{R}^n$
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Among all subsets $A\subset \mathbb{R}^n$ with a given volume, the Euclidean ball has the minimal area.
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That is, for any $\epsilon>0$, Euclidean balls minimize the volume of the $\epsilon$-neighborhood of $A$.
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Where the volume of the $\epsilon$-neighborhood of $A$ is defined as
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$$
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A_\epsilon(A)\coloneqq \{x\in \mathbb{R}^n: \exists y\in A, \|x-y\|_2\leq \epsilon\}=A+\epsilon B_2^n
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$$
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Here the $\|\cdot\|_2$ is the Euclidean norm. (The theorem holds for both geodesic metric on sphere and Euclidean metric on $\mathbb{R}^n$.)
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#### Isoperimetric inequality on the sphere
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Let $\sigma_n(A)$ denotes the normalized area of $A$ on $n$ dimensional sphere $S^n$. That is $\sigma_n(A)\coloneqq\frac{\operatorname{Area}(A)}{\operatorname{Area}(S^n)}$.
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Let $\epsilon>0$. Then for any subset $A\subset S^n$, given the area $\sigma_n(A)$, the spherical caps minimize the volume of the $\epsilon$-neighborhood of $A$.
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> The above two inequalities is not proved in the Book _High-dimensional probability_. But you can find it in the Appendix C of Gromov's book _Metric Structures for Riemannian and Non-Riemannian Spaces_.
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To continue prove the theorem, we use sub-Gaussian concentration *(Chapter 3 of _High-dimensional probability_ by Roman Vershynin)* of sphere $\sqrt{n}S^n$.
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This will leads to some constant $C>0$ such that the following lemma holds:
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#### The "Blow-up" lemma
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Let $A$ be a subset of sphere $\sqrt{n}S^n$, and $\sigma$ denotes the normalized area of $A$. Then if $\sigma\geq \frac{1}{2}$, then for every $t\geq 0$,
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$$
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\sigma(A_t)\geq 1-2\exp(-ct^2)
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$$
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where $A_t=\{x\in S^n: \operatorname{dist}(x,A)\leq t\}$ and $c$ is some positive constant.
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#### Proof of the Levy's concentration theorem
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Proof:
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Without loss of generality, we can assume that $\eta=1$. Let $M$ denotes the median of $f(X)$.
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So $\operatorname{Pr}[|f(X)\leq M|]\geq \frac{1}{2}$, and $\operatorname{Pr}[|f(X)\geq M|]\geq \frac{1}{2}$.
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Consider the sub-level set $A\coloneqq \{x\in \sqrt{n}S^n: |f(x)|\leq M\}$.
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Since $\operatorname{Pr}[X\in A]\geq \frac{1}{2}$, by the blow-up lemma, we have
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$$
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\operatorname{Pr}[X\in A_t]\geq 1-2\exp(-ct^2)
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$$
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And since
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$$
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\operatorname{Pr}[X\in A_t]\leq \operatorname{Pr}[f(X)\leq M+t]
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$$
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Combining the above two inequalities, we have
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$$
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\operatorname{Pr}[f(X)\leq M+t]\geq 1-2\exp(-ct^2)
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$$
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## Levy's concentration theorem in _Metric Structures for Riemannian and Non-Riemannian Spaces_ by M. Gromov
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### Levy's concentration theorem (Gromov's version)
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> The Levy's lemma can also be found in _Metric Structures for Riemannian and Non-Riemannian Spaces_ by M. Gromov. $3\frac{1}{2}.19$ The Levy concentration theory.
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#### Theorem $3\frac{1}{2}.19$ Levy concentration theorem:
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An arbitrary 1-Lipschitz function $f:S^n\to \mathbb{R}$ concentrates near a single value $a_0\in \mathbb{R}$ as strongly as the distance function does.
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That is
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$$
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\mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp(-\frac{(n-1)\epsilon^2}{2})
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$$
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where
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$$
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\kappa_n(\epsilon)=\frac{\int_\epsilon^{\frac{\pi}{2}}\cos^{n-1}(t)dt}{\int_0^{\frac{\pi}{2}}\cos^{n-1}(t)dt}
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$$
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$a_0$ is the **Levy mean** of function $f$, that is the level set of $f^{-1}:\mathbb{R}\to S^n$ divides the sphere into equal halves, characterized by the following equality:
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$$
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\mu(f^{-1}(-\infty,a_0])\geq \frac{1}{2} \text{ and } \mu(f^{-1}[a_0,\infty))\geq \frac{1}{2}
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$$
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Hardcore computing may generates the bound but M. Gromov did not make the detailed explanation here.
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> Detailed proof by Takashi Shioya.
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>
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> The central idea is to draw the connection between the given three topological spaces, $S^{2n+1}$, $CP^n$ and $\mathbb{R}$.
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First, we need to introduce the following distribution and lemmas/theorems:
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**OBSERVATION**
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consider the orthogonal projection from $\mathbb{R}^{n+1}$, the space where $S^n$ is embedded, to $\mathbb{R}^k$, we denote the restriction of the projection as $\pi_{n,k}:S^n(\sqrt{n})\to \mathbb{R}^k$. Note that $\pi_{n,k}$ is a 1-Lipschitz function (projection will never increase the distance between two points).
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We denote the normalized Riemannian volume measure on $S^n(\sqrt{n})$ as $\sigma^n(\cdot)$, and $\sigma^n(S^n(\sqrt{n}))=1$.
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#### Definition of Gaussian measure on $\mathbb{R}^k$
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We denote the Gaussian measure on $\mathbb{R}^k$ as $\gamma^k$.
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$$
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d\gamma^k(x)\coloneqq\frac{1}{\sqrt{2\pi}^k}\exp(-\frac{1}{2}\|x\|^2)dx
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$$
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$x\in \mathbb{R}^k$, $\|x\|^2=\sum_{i=1}^k x_i^2$ is the Euclidean norm, and $dx$ is the Lebesgue measure on $\mathbb{R}^k$.
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Basically, you can consider the Gaussian measure as the normalized Lebesgue measure on $\mathbb{R}^k$ with standard deviation $1$.
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#### Maxwell-Boltzmann distribution law
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> It is such a wonderful fact for me, that the projection of $n+1$ dimensional sphere with radius $\sqrt{n}$ to $\mathbb{R}^k$ is a Gaussian distribution as $n\to \infty$.
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For any natural number $k$,
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$$
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\frac{d(\pi_{n,k})_*\sigma^n(x)}{dx}\to \frac{d\gamma^k(x)}{dx}
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$$
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where $(\pi_{n,k})_*\sigma^n$ is the push-forward measure of $\sigma^n$ by $\pi_{n,k}$.
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In other words,
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$$
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(\pi_{n,k})_*\sigma^n\to \gamma^k\text{ weakly as }n\to \infty
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$$
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<details>
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<summary>Proof</summary>
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We denote the $n$ dimensional volume measure on $\mathbb{R}^k$ as $\operatorname{vol}_k$.
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Observe that $\pi_{n,k}^{-1}(x),x\in \mathbb{R}^k$ is isometric to $S^{n-k}(\sqrt{n-\|x\|^2})$, that is, for any $x\in \mathbb{R}^k$, $\pi_{n,k}^{-1}(x)$ is a sphere with radius $\sqrt{n-\|x\|^2}$ (by the definition of $\pi_{n,k}$).
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So,
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$$
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\begin{aligned}
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\frac{d(\pi_{n,k})_*\sigma^n(x)}{dx}&=\frac{\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(x))}{\operatorname{vol}_k(S^n(\sqrt{n}))}\\
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&=\frac{(n-\|x\|^2)^{\frac{n-k}{2}}}{\int_{\|x\|\leq \sqrt{n}}(n-\|x\|^2)^{\frac{n-k}{2}}dx}\\
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\end{aligned}
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$$
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as $n\to \infty$.
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note that $\lim_{n\to \infty}{1-\frac{a}{n}}=e^{-a}$ for any $a>0$.
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$(n-\|x\|^2)^{\frac{n-k}{2}}=\left(n(1-\frac{\|x\|^2}{n})\right)^{\frac{n-k}{2}}\to n^{\frac{n-k}{2}}\exp(-\frac{\|x\|^2}{2})$
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So
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$$
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\begin{aligned}
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\frac{(n-\|x\|^2)^{\frac{n-k}{2}}}{\int_{\|x\|\leq \sqrt{n}}(n-\|x\|^2)^{\frac{n-k}{2}}dx}&=\frac{e^{-\frac{\|x\|^2}{2}}}{\int_{x\in \mathbb{R}^k}e^{-\frac{\|x\|^2}{2}}dx}\\
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&=\frac{1}{(2\pi)^{\frac{k}{2}}}e^{-\frac{\|x\|^2}{2}}\\
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&=\frac{d\gamma^k(x)}{dx}
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\end{aligned}
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$$
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QED
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</details>
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#### Proof of the Levy's concentration theorem via the Maxwell-Boltzmann distribution law
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We use the Maxwell-Boltzmann distribution law and Levy's isoperimetric inequality to prove the Levy's concentration theorem.
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The goal is the same as the Gromov's version, first we bound the probability of the sub-level set of $f$ by the $\kappa_n(\epsilon)$ function by Levy's isoperimetric inequality. Then we claim that the $\kappa_n(\epsilon)$ function is bounded by the Gaussian distribution.
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<details>
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<summary>Proof</summary>
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Let $f:S^n\to \mathbb{R}$ be a 1-Lipschitz function.
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We define $\kappa_n(\epsilon)$ as the following:
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$$
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\kappa_n(\epsilon)=\frac{\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\epsilon))}{\operatorname{vol}_k(S^n(\sqrt{n}))}=\frac{\int_\epsilon^{\frac{\pi}{2}}\cos^{n-1}(t)dt}{\int_0^{\frac{\pi}{2}}\cos^{n-1}(t)dt}
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$$
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By the Levy's isoperimetric inequality, and the Maxwell-Boltzmann distribution law, we have
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$$
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\mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp(-\frac{(n-1)\epsilon^2}{2})
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$$
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</details>
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## Levy's Isoperimetric inequality
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> This section is from the Appendix $C_+$ of Gromov's book _Metric Structures for Riemannian and Non-Riemannian Spaces_.
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Not very edible for undergraduates.
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### Riemannian manifolds
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## References
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- [High-dimensional probability by Roman Vershynin](https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-2.pdf)
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- [Metric Structures for Riemannian and Non-Riemannian Spaces by M. Gromov](https://www.amazon.com/Structures-Riemannian-Non-Riemannian-Progress-Mathematics/dp/0817638989/ref=tmm_hrd_swatch_0?_encoding=UTF8&dib_tag=se&dib=eyJ2IjoiMSJ9.Tp8dXvGbTj_D53OXtGj_qOdqgCgbP8GKwz4XaA1xA5PGjHj071QN20LucGBJIEps.9xhBE0WNB0cpMfODY5Qbc3gzuqHnRmq6WZI_NnIJTvc&qid=1750973893&sr=8-1)
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- [Metric Measure Geometry by Takashi Shioya](https://arxiv.org/pdf/1410.0428) |