update

content/Math401/Math401_P1_1.md

@@ -142,3 +142,7 @@ is a pure state.
 
 QED
 </details>
+
+## Drawing the connection between the space $S^{2n+1}$, $CP^n$, and $\mathbb{R}$
+
+##

content/Math401/Math401_P1_2.md

@@ -66,7 +66,9 @@ The Haar measure is the unique probability measure that is invariant under the a
 
 _The existence and uniqueness of the Haar measure is a theorem in compact lie group theory. For this research topic, we will not prove it._
 
-### Sub-Gaussian concentration
+### Maxwell-Boltzmann distribution and projection of high-dimensional sphere
+
+
 
 ### Random sampling on the $CP^n$
 
@@ -90,9 +92,6 @@ $$
 S_{m,n}=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n}\simeq \ln m-\frac{m}{2n}
 $$
 
-
-
-
 ## References
 
 - [The random Matrix Theory of the Classical Compact groups](https://case.edu/artsci/math/esmeckes/Haar_book.pdf)

content/Math401/Math401_P1_3.md

@@ -42,7 +42,7 @@ Let $\sigma_n(A)$ denotes the normalized area of $A$ on $n$ dimensional sphere $
 
 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 $A$.
 
-The above two inequalities is not proved in the Book _High-dimensional probability_.
+> 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_.
 
 To continue prove the theorem, we use sub-Gaussian concentration *(Chapter 3 of _High-dimensional probability_ by Roman Vershynin)* of sphere $\sqrt{n}S^n$. 
 
@@ -116,7 +116,7 @@ $$
 
 Hardcore computing may generates the bound but M. Gromov did not make the detailed explanation here.
 
-> Detail proof by Takashi Shioya.
+> Detailed proof by Takashi Shioya.
 >
 > The central idea is to draw the connection between the given three topological spaces, $S^{2n+1}$, $CP^n$ and $\mathbb{R}$.
 
@@ -176,10 +176,58 @@ $$
 
 as $n\to \infty$.
 
+note that $\lim_{n\to \infty}{1-\frac{a}{n}}=e^{-a}$ for any $a>0$.
+
 $(n-\|x\|^2)^{\frac{n-k}{2}}=\left(n(1-\frac{\|x\|^2}{n})\right)^{\frac{n-k}{2}}\to n^{\frac{n-k}{2}}\exp(-\frac{\|x\|^2}{2})$
 
+So
+
+$$
+\begin{aligned}
+\frac{(n-\|x\|^2)^{\frac{n-k}{2}}}{\int_{\|x\|\leq \sqrt{n}}(n-\|x\|^2)^{\frac{n-k}{2}}dx}&=\frac{e^{-\frac{\|x\|^2}{2}}}{\int_{x\in \mathbb{R}^k}e^{-\frac{\|x\|^2}{2}}dx}\\
+&=\frac{1}{(2\pi)^{\frac{k}{2}}}e^{-\frac{\|x\|^2}{2}}\\
+&=\frac{d\gamma^k(x)}{dx}
+\end{aligned}
+$$
+
+QED
+
 </details>
 
+#### Proof of the Levy's concentration theorem via the Maxwell-Boltzmann distribution law
+
+We use the Maxwell-Boltzmann distribution law and Levy's isoperimetric inequality to prove the Levy's concentration theorem.
+
+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.
+
+<details>
+<summary>Proof</summary>
+
+Let $f:S^n\to \mathbb{R}$ be a 1-Lipschitz function.
+
+We define $\kappa_n(\epsilon)$ as the following:
+
+$$
+\kappa_n(\epsilon)=\frac{\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\epsilon))}{\operatorname{vol}_k(S^n(\sqrt{n}))}=\frac{\int_\epsilon^{\frac{\pi}{2}}\cos^{n-1}(t)dt}{\int_0^{\frac{\pi}{2}}\cos^{n-1}(t)dt}
+$$
+
+By the Levy's isoperimetric inequality, and the Maxwell-Boltzmann distribution law, we have
+
+$$
+\mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp(-\frac{(n-1)\epsilon^2}{2})
+$$
+</details>
+
+## Levy's Isoperimetric inequality
+
+> This section is from the Appendix $C_+$ of Gromov's book _Metric Structures for Riemannian and Non-Riemannian Spaces_.
+
+Not very edible for undergraduates.
+
+### Riemannian manifolds
+
+
+
 ## References
 
 - [High-dimensional probability by Roman Vershynin](https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-2.pdf)