diff --git a/chapters/chap2.pdf b/chapters/chap2.pdf
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diff --git a/chapters/chap2.tex b/chapters/chap2.tex
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--- a/chapters/chap2.tex
+++ b/chapters/chap2.tex
@@ -109,13 +109,25 @@ Few additional proposition in \cite{shioya2014metricmeasuregeometry} will help u
   Since $f$ is 1-Lipschitz, we have $f_*\mu Y=\mu_X$. Let $A$ be any Borel set of $Y$ with $\mu_Y(A)\geq 1-\kappa$ and $\overline{f(A)}$ be the closure of $f(A)$ in $X$. We have $\mu_X(\overline{f(A)})=\mu_Y(f^{-1}(\overline{f(A)}))\geq \mu_Y(A)\geq 1-\kappa$ and by the 1-lipschitz property, $\diam(\overline{f(A)})\leq \diam(A)$, so $\diam(X;1-\kappa)\leq \diam(A)\leq \diam(Y;1-\kappa)$.
 
   Let $g:X\to \R$ be any 1-lipschitz function, since $(\R,|\cdot|,g_*\mu_X)$ is dominated by $X$, $\diam(\R;1-\kappa)\leq \diam(X;1-\kappa)$. Therefore, $\obdiam(X;-\kappa)\leq \diam(X;1-\kappa)$.
-
+ 
   and
   $$
   \diam(g_*\mu_X;1-\kappa)\leq \diam((f\circ g)_*\mu_Y;1-\kappa)\leq \obdiam(Y;1-\kappa)
   $$
 \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
+
+\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.
+
+\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.
+
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