update

chapters/chap2.tex

@@ -90,6 +90,7 @@ The point of the $\max\{\cdot,\kappa\}$ is that it prevents cheating by taking $
 Few additional proposition in \cite{shioya2014metricmeasuregeometry} will help us to estimate the observable diameter for complex projective spaces.
 
 \begin{prop}
+  \label{prop:observable-diameter-domination}
   Let $X$ and $Y$ be two metric-measure spaces and $\kappa>0$, and let $f:Y\to X$ be a 1-Lipschitz function ($Y$ dominates $X$, denoted as $X\prec Y$) then:
 
   \begin{enumerate}
@@ -100,7 +101,7 @@ Few additional proposition in \cite{shioya2014metricmeasuregeometry} will help u
     \item $\obdiam(X;-\kappa)\leq \diam(X;1-\kappa)$, and $\obdiam(X)$ is finite.
     \item
     $$
-    \obdiam(X;-kappa)\leq \obdiam(Y;-kappa)
+    \obdiam(X;-\kappa)\leq \obdiam(Y;-\kappa)
     $$
   \end{enumerate}
 \end{prop}
@@ -116,14 +117,81 @@ Few additional proposition in \cite{shioya2014metricmeasuregeometry} will help u
   $$
 \end{proof}
 
+\begin{prop}
+  \label{prop:observable-diameter-scale}
+  Let $X$ be an metric-measure space. Then for any real number $t>0$, we have
+  
+  $$
+  \obdiam(tX;-\kappa)=t\obdiam(X;-\kappa)
+  $$
+
+  Where $tX=(X,tdX,\mu X)$.
+\end{prop}
+
+\begin{proof}
+  $$
+  \begin{aligned}
+    \obdiam(tX;-\kappa)&=\sup\{\diam(f_*\mu_X;1-\kappa)|f:tX\to \R \text{ is 1-Lipschitz}\}\\
+    &=\sup\{\diam(f_*\mu_X;1-\kappa)|t^{-1}f:X\to \R \text{ is 1-Lipschitz}\}\\
+    &=\sup\{\diam((tg)_*\mu_X;1-\kappa)|g:X\to \R \text{ is 1-Lipschitz}\}\\
+    &=t\sup\{\diam(g_*\mu_X;1-\kappa)|g:X\to \R \text{ is 1-Lipschitz}\}\\
+    &=t\obdiam(X;-\kappa)
+  \end{aligned}
+  $$
+\end{proof}
+
 \subsection{Observable diameter for class of spheres}
 
-In this section, we will try to use the results from previous sections to estimate the observable diameter for class of spheres
+In this section, we will try to use the results from previous sections to estimate the observable diameter for class of spheres.
+
+\begin{theorem}
+  \label{thm:observable-diameter-sphere}
+  For any real number $\kappa$ with $0<\kappa<1$, we have
+  $$
+  \obdiam(S^n(1);-\kappa)=O(\sqrt{n})
+  $$
+\end{theorem}
+
+\begin{proof}
+  First, recall that by maxwell boltzmann distribution, we have that for any $n>0$, let $I(r)$ denote the measure of standard gaussian measure on the interval $[0,r]$. Then we have
+  
+  $$
+  \begin{aligned}
+    \lim_{n\to \infty} \obdiam(S^n(\sqrt{n});-\kappa)&=\lim_{n\to \infty} \sup\{\diam((\pi_{n,k})_*\sigma^n;1-\kappa)|\pi_{n,k} \text{ is 1-Lipschitz}\}\\
+    &=\lim_{n\to \infty} \sup\{\diam(\gamma^1;1-\kappa)|\gamma^1 \text{ is the standard gaussian measure}\}\\
+    &=\diam(\gamma^1;1-\kappa)\\
+    &=2I^{-1}(\frac{1-\kappa}{2})\text { cutting the extremum for normal distribution}\\
+  \end{aligned}
+  $$
+
+  By proposition \ref{prop:observable-diameter-scale}, we have
+
+  $$
+  \obdiam(S^n(\sqrt{n});-\kappa)=\sqrt{n}\obdiam(S^n(1);-\kappa)
+  $$
+
+  So $\obdiam(S^n(1);-\kappa)=\sqrt{n}(2I^{-1}(\frac{1-\kappa}{2}))=O(\sqrt{n})$.
+\end{proof}
 
 \subsection{Observable diameter for complex projective spaces}
 
 Using the projection map and Hopf's fibration, we can estimate the observable diameter for complex projective spaces from the observable diameter of spheres.
 
+\begin{theorem}
+  \label{thm:observable-diameter-complex-projective-space}
+  For any real number $\kappa$ with $0<\kappa<1$, we have
+  $$
+  \obdiam(\mathbb{C}P^n(1);-\kappa)\leq O(\sqrt{n})
+  $$
+\end{theorem}
+
+\begin{proof}
+  Recall from Example 2.30 in \cite{lee_introduction_2018}, the Hopf fibration $f_n:S^{2n+1}(1)\to \C P^n$ is 1-Lipschitz continuous with respect to the Fubini-Study metric on $\C P^n$. and the push-forward $(f_n)_*\sigma^{2n+1}$ coincides with the normalized volume measure on $\C P^n$ induced from the Fubini-Study metric.
+
+  By proposition \ref{prop:observable-diameter-domination}, we have $\obdiam(\mathbb{C}P^n(1);-\kappa)\leq \obdiam(S^{2n+1}(1);-\kappa)\leq O(\sqrt{n})$.
+
+\end{proof}
+
 \subsection{More example for concentration of measure and observable diameter}
 
 In this section, we wish to use observable diameter to estimate the statics of thermal dynamics of some classical systems.

main.bib

@@ -84,6 +84,17 @@
       url={https://arxiv.org/abs/1410.0428}, 
 }
 
+@book{lee_introduction_2018,
+  title = {Introduction to {{Riemannian Manifolds}}},
+  author = {Lee, John M.},
+  year = {2018},
+  series = {Graduate {{Texts}} in {{Mathematics}}},
+  edition = {2nd},
+  publisher = {Springer},
+  address = {Cham, Switzerland},
+  isbn = {978-3-319-91755-9}
+}
+
 @inproceedings{Hayden,
   title     = {Concentration of measure effects in quantum information},
   author    = {Hayden, Patrick},